Those are the rules I use for my Chtulhu-esque investigation games. I like BitD, so it's inspired by that system.

Rules:

Action roll:

Action-rolls are the mechanism we use to resolve conflicts in this game. When the

Investigators want to do something and there is an interesting

obstacle between them and what they want, they roll a number of d6 equal to

the rating of the Action they use to overcome the obstacle; in practice that's a couple of d6.

Then we look for the highest number rolled.

If the highest die of the roll shows a 3, the Investigator fails - they should be ready

for the worst (in RPG-speak: 'the GM gets to make a move'). Something goes wrong in some way or another and the GM can introduce a new complication AND the obstacle isn't overcome. If the highest die shows a 4 or 5, the Investigator gets what they want and they overcome the obstacle, however, they will have to pay a price to do so(in RPG-speak: 'the GM gets to make a move' also). If the highest die shows a 6 they succeed without repercussions. If the dice show more than one 6, they critically succeed and get +1 Effect. More on Effect further down. The idea is that the GM never rolls any dice but simply moves when the Investigators ignore an obvious danger or when they get no 6 on an Action-roll. For rolling, the Investigator:

Takes 1d6 per rating of the Action they are using.
Takes +1d6 or +1 Effect if they get any help at all. Everyone helping is in the line of fire for consequences.
Takes +1d6 or +1 Effect per significant advantage ( great tools, great plan etc. )

If the Investigator has 0 dice left after accumulating a dice pool, they roll 2d6 but look at the worse of the two to evaluate the roll.

Aside from whether the Action succeeds, another factor is how effective the Action is in overcoming an obstacle. That's where 'Effect' comes in. Every Action has an Effect of 1 and is modified by the schema above and by Perks. Actions that are very unlikely to succeed, either because of narrative disadvantages or because an approach was chosen unwisely can have -Effect. This means that the Investigators will have to make do with less than what they need to fully overcome an obstacle. Even Actions with an Effect of 0 should have some effect, it's just very limited in scope.

A useful way to think about Effect and more complicated obstacles is to treat some obstacles as extended tests on which Effect accumulates until the obstacle is overcome. Nowadays, extended tests are called 'Clocks', the syntax being that a 6-Clock requires a total Effect of 6, from different Actions, to overcome. For instance, if escaping the cultists is a 6-Clock, Actions with a total Effect of 6 have to be rolled to escape them. For more complicated obstacles, writing the Clocks down is helpful, though, just keeping Clocks in the back of the head for smaller situations is enough.

As a general rule of thumb: Whenever an Investigator got more Effect than they need they should get some extra benefit for every point of Effect. That means, if they got +Effect and it's just a regular roll where we don't need Clocks they can ask for extra for these rolls.

Health & Will

Whenever an Investigator suffers mental or physical harm they have to roll a number of d6 =

to their Will or Health stat - depending on the nature of the suffering.

They reduce their number of dice by one if the suffering is particularly severe,

by two if it is life-threatening, and if it is caused by the mythos by three.

The system for Health or Will checks works like the-price-is-right,

when suffering harm an Investigator rolls a number of d6 and attempts to get

as close as possible to their rating of Health or Will respectively. After the roll, the

Investigators then reduce their Health or Will stat by a number equal

to the difference between their current stat (NOT the number of dice rolled)

and the closest number rolled. Whenever an Investigator loses Health or Will they are narratively injured and the GM can ask them to also use the Endure Action if they want to do stuff their injury hinders them. Armour or cover gives extra dice on Health tests.

So, for instance: Alex gets shot by a small gun. They try to dodge but fail.

Alex's Health stat is currently 4, because they were already injured. It's a gun, so

Alex is told to reduce their pool by 1; 3 dice left. Their dice show: 1,1,6. A bad roll.

The closest number to their Health (which was 4) is 6, so they reduce their health by the

difference of 4 and 6 which is 2! Now Alex's Health is 2 and because they got an injury, the GM can ask them to make an Action roll to Endure the wound when they do things that are too straining.

If after a Will roll the stat is 0, the Investigator is about to break,
and if they break they suffer Trauma. While at 0 Will, any Investigator that acts in
a meaningful capacity also has to use the Endure Action. If they fail to Endure the pain
and hardship they get Trauma.
Suffering Trauma hands the GM control of the PC for the scene: they are antagonistic,
knocked out from pain, temporally insane, or otherwise unhelpful. Every time an
Investigator suffers Trauma their friends suffer with them and the traumatized
Investigator reduces one friend's rating by 1d4. When a friend's rating reaches 0 they are gone
and with no friends left Investigators become unplayable. An Investigator will reappear in the
next appropriate scene, with Will back at 1d6.
On the other hand, the body is less resilient than the mind. Health works the same
as Will except that Investigators die instead of suffering Trauma. Whenever an Investigator
is at 0 Health and they suffer harm again, they are dying. At 0, they are barely stable and
have to Endure to act. While dying they need outside assistance (in the form of an Action Roll)

or, well, they die.

'Push-Yourself': Investigators can choose to push themselves. To do so, after any Action-roll but only once per roll they can reduce their current Health or Will by 1 and reroll any of the dice.

Investigator Creation:

Actions ratings:

The Action ratings of the Investigators are the following. Importantly: they are verbs, things the Investigators do (to overcome obstacles), not attributes they have. People can be good at noticing things for all kinds of reasons but we don't really care about why they are good in this game. If you disagree with me on this, you can easily select a different number of Action or differently selected or worded Actions. (Though you might have to fiddle a little with the numbers).

The Actions should be somewhat self-explanatory.

The Investigators have two Actions at 0, four Actions at 1, four Actions at 2, and two Actions at 3.

Notice (awareness, hearing, reflexes, searching)
Assess(interpretation, estimation, insight, analysis)
Move (Moving oneself and or objects from place A to place B, including sprinting, climbing etc.)
Charm (Persuasion; being nice and chummy with people)
Coerce (Intimidating, commanding, and shouting)
Deceive (Lying, trickery, disguise, misdirection)
Procure (Buying things, finding the right stores, places, information and people)
Fix (Making things work again, machines or biological)
Fight (Fisticuffs and melee weapons)
Shoot (Shooting)
Stealth (Being inconspicuous, evading notice, hiding things)
Endure (Enduring uncomfortable conditions before or after harm occurs)

Stats: All Investigators start with the following statistics, though some of them can be customized further through their Background.

Health 6
Will 6
Influence 3
Preparation 7

Influence:

Investigators can create a number of friends, i.e. of people who are willing to help and get in danger for them, equal to their Influence stat.

When creating a friend first assign them a Name and a Background. Their Background provides their expertise, the avenue through which they can help the Investigators. Every friend stars with a rating of 4, asking them for favours reduces the rating. As a rule of thumb, to get stuff from friends one needs to reduce their rating by 1.

It's fine to create friends on the fly.

Preparation:

Investigators have stuff. However, I find tracking that kind of stuff and going on shopping sprees unimaginably boring. To simulate their planning and the stuff they 'always already had' on them (they can find more things during play, of course) I use the Preparation stat.

During play, every Investigator can 'create' a number of items they already had on them equal to their Preparation stat. Investigators have to make up where they got the item from, they can't just generate a shotgun from nowhere but the shotgun might be in their car. Which items require to spent Preparation is up to the GM, however, to buy stuff we use the Procure Action. Points of their Preparation that were spend refresh every Investigation.

Calculating the cost to have an item prepared goes as follows:

Size:

Small: 1 (handheld)

Large: 4 (larger than that)

Unlikely: +1 - +3 (just something that isn't likely for the Investigator to have, +1 would be something unlikely, like a fishing rod in a city and +3 is something highly specific, like an invitation to a particular party. If it's too out there the Investigator can't have it.)

Illegal: +1

Expensive: +1

Fine Quality: +3 (gives +1d or +1 Effect)

Equipment opens up the possibility to do things: you can't shoot a gun if you don't have one. Items of a Fine Quality give a bonus, otherwise, small weapons give +1 Effect and large Weapons give +2 Effect. Armour, such as a bulletproof vest, counts as large and gives +1d on Health tests when it applies.

Background:

Every Investigator has a Background. Backgrounds are the Investigator's day job or what they learned to do. They get 13 points:

(1/3) Outdoors-Kid: For 1 Point, the Investigator can always find North, for 3 points the Investigator can survive outdoors for an infinite amount of time.
(1/2/3) Good Reputation: The Investigator gets 3 points of Reputation that can be spent to get +1d on social Actions they do that relate to what people might have heard about them. This pool refreshes every session.
(1) Linguist: The Investigator can speak an additional language. Can be taken multiple times.
(2/4) First Aid Expert: Once/twice per session, the Investigator can attempt to temporarily Fix the Health of another Investigator. The target heals an amount of Health = the Effect of the Action. After the investigation, the injury reappears and the Health drops to a minimum of 0.
(2/4) IT-Expert: The Investigator gets +1 Effect when interfacing with a computer. For 4 points they also get +1d when doing so.
(2/4) Silent: The Investigator gets +1 Effect to avoid being heard. For 4 points they also get +1d.
(2/4) Veteran: The Investigator's Will does not suffer from regular death scenes. For 4 points they can even endure gruesome scenes of carnage.
(2/4/6) Educated: 1/2/3 times per Session, the Investigator can provide a useful bit of information from their field of education (as determined by the Background). Whenever people use that information for action, they get +1d (does not stack).
(2) Well prepared: The investigator starts with +1 Preparation. Can be taken up to 5 times.
(2) Criminal: The Investigator has easy access to the criminal underground of the area. When rolling to procure illegal goods, take +1 Effect.
(2) Contortionist: The Investigator is able to crawl through small spaces and contort themselves in ways not thought possible.
(2) Secret Identity: The Investigator gets +1d whenever they try to hide their true identity.
(2) Academic: The Investigator chooses a field of study and gets +1d to know things in relation to that field.
(2) Innocent-looking: +1 Effect on rolls to de-escalate situations.
(2) Iron stomach: The Investigator can eat lots of things that might make other folks sick and gets +1d to Endure poisons and +1d on Health tests vs poison.
(2) Athletic: The Investigator has +1 Effect on rolls to run or jump and on stamina-related tasks.
(2) Juryrigger: The Investigator can make a machine, that's normally just done with the world, and continue until the end of the scene.
(2) Hawk-eyes: The Investigator's eyes have a regular Effect on longer distances.
(2) Engineer: The Investigator has to select a speciality for the Fix Action. When the Investigator Fixes their subject they get +1 Effect.
(2) Book Worm: The Investigator gets +1d to research in a library.
(2) Animal-lover: The Investigator can practically speak with animals (not really but almost) and animals don't like to hurt them.
(2) Lucky: The Investigator gets 1 use per session, expending a use after any roll, they can roll an additional d6. Only once per roll and can be taken multiple times.
(3/6) Trained: The Investigator gets +1 Effect for combat. For 6 points, they also get +1d.
(3) Scary: Take +1d to scare or intimidate people.
(3) Official: The Investigator gets +1d to maneuver bureaucracies.
(3) Hotshot Driver: +1d to drive vehicles.
(3) Likeable: The Investigator starts with an additional friend.
(3) People Person: The investigator's friends start with an additional rating point.
(3) Reflexes: When reaction speed is in question, the Investigator gets to react first.
(3) Sweetheart: The Investigator takes +1d on all actions against people they had some intimacy with.
(3) Busy-body: The Investigator has +1 Downtime Action.
(3) Night-owl: The Investigator can see pretty well in (not totally black) darkness and you are good with little sleep.
(3) Sucker for Pain: The Investigator gets +1d to endure pain.
(3) Occultist: The Investigator gets +1d to study and use dangerous tomes.
(3) Anchored: Once per session, the Investigator can roll a Will Test again.
(3) Feline Grace: The Investigator gets +1d to keep balance.
(3) Scarred: The Investigator starts with knowledge of the Mythos and can resist with Will to recall Information about it. Additionally, they have -1 max. Will but take +1d when casting spells or when resisting the Mythos with Will.
(3) Well-Connected: The Investigator gets +1d to procure people.
(3) Fast Recovery: The Investigator gets +1 Effect when using the Recover Downtime-Action.
(3) Nimble: The Investigator gets +1d for tasks involving sleight of hands, like magic (the non-spellcasting kind) or pickpocketing.
(3) Martial Arts: The Investigator takes +1d when they fight without attempting to deal damage (like, when they try to disarm, grapple, knock someone prone etc.)
(4) Tough-skinned: The Investigator takes +1d on Health rolls. (This doesn't change the stat they need to hit with their roll.)
(4) Iron-willed: The Investigator takes +1d on Will rolls. (This doesn't change the stat they need to hit with their roll.)
(4) Well-Off: The Investigator has +1 on Preparation and +1d to procure things with money.
(4) Photographic Memory: It does what it says.
(4) Strong: The Investigator has +1 Effect on Move and Fight Actions that involve muscle power.
(4) Charismatic: Chose one Action from either Charm, Deceive or Coerce. The Investigator gets +1 Effect whenever they use the Action.
(4) Talented: The Investigator gets an additional point to distribute on their Actions.
(4) Wing Man: Help of the Investigator gives +1d as well as +1 Effect (instead of only one).
(5) Assistant: The Investigator has another PC that helps them. The Assistant has all Actions on 0, except one which is on 2.
(6) Medium: The Investigator has a supernatural sense to notice Mythos activity.
(6) Trained to Kill: The Investigator takes +3 Effect on all Actions that attempt to murder someone.
(7) Sixth Sense: The Investigator can always tell when someone is lying (i.e. when someone is saying something they know is not true, in order to deceive).

Downtime:

After every investigation, the Investigators can take three Downtime-Actions. During downtime, rolls work a little differently, but every Downtime-Action requires a roll. Here: 1-3 = 1 Effect, 4-5 = 2 Effect, 6 = 3 Effect and +1 Effect per additional 6.

Basically, we fill Clocks during downtime. The following Clocks are provided by me but the GM can make up new ones depending on what the Investigators want to do in the fiction.

Recover: Regain Health or Will by the Effect of the roll to a maximum of 6.
Improve: 6-Clock. The rating of an Action increases by 1.
Hang out: 6-Clock. The rating of a friend increases by 1d4.
Work and make some money: 4-Clock. The Preparation stat increases by 1.
Research: Ask the GM a question you want to answer through research. The clock is relative to the question: 2-Clock = normal, 4-Clock = difficult. 6-Clock = very difficult.
Bond: Roll a Downtime Action for another Investigator.
School: 6-Clock. Get 1d4 Background points.
Reach out: 8-Clock. Get another friend with a rating of 1d4.

Furthermore: there might be Clocks already in play from the investigations. Those can stay in play and the Investigators can influence them by Spending Downtime-Actions also. For that, they have to say how they do it and then roll an Action. They change the Clock by their Effect as with other Downtime-Actions.

I suggest not just playing out any downtime abstractly but laying plot-hooks and opportunities for Action-Rolls, if the Investigators have the opportunity to find some stuff out during the fiction: let them.

Magick:

As in other Chtulhu-esque games, there are ways to engage the Mythos as Investigators that does not involve running, hiding or shooting. Some cultists might have access to Magick and when finding dangerous tomes the characters might also get access to not only knowledge but also spells. In general, I found it best not to make Magick too accessible but that's just the way I run my games. I find it takes away from the weirdness when you can control it with dice rolls.

In case the Investigators get hold of dark tomes and are able to learn some spells, the best way I found how to handle them is idiosyncratic. Every tome should have its own rules, dice rolls, and every spell should work slightly differently from all others. A general scheme, which should be filled out more dutifully by the GM, could look like so:

Reading tomes requires an Action-roll. Most likely, Asses, Endure, and Procure; but others are thinkable depending on the work. Maybe the Investigator has to softly whisper charming words so the book opens up to them? Afterwards, there should be some kind of Will roll with a modifier depending on how bad the tome is. Knowledge in these games is dangerous. Having read the tome should always give a narrative benefit: now the Investigator knows something relevant they didn't before. Maybe the weakness of a horror or maybe a new clue where to go next.

Tomes could include spells. I think, learning a spell really depends on the spell and the tome. If it's a quick chant maybe the Investigator can learn it just while studying the tome, with no further roll needed but maybe it's a hidden ritual that has to be uncovered with more rolls.

Spells that are uncovered can be used (who would have guessed). Spells should be handled individually also. There are different axis that can be modified to create an individual spell.

Magick: What does the spell actually do? I think the best spells have effects that are explainable otherwise. Only powerful Magick should be visibly magical.

Action: Does the spell require an Action to cast or does it just work?

Will: Whenever an Investigator casts a spell they require to test with Will. Casting spells destroys the mind. However, powerful Magick might even remove some dice, max. 3.

Health: In addition, some Magick might require a Health test since it drains the life of the caster.

Time: How long does casting the spell take? A few seconds, hours, days? Is it a ritual?

Ingredients: Does the spell require special preparation? Some ingredients are easy to gather but for some, the Investigators might need to spend time, money and effort to gather.

Memorization: Can the spell be learned by heart or does the caster require notes of the incantation?

I know this is not incredibly helpful, though, I think the best spells are just designed by the GM, for the book it can be found in and for a specific investigation or campaign.

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Arkham Stories (I.e. the Cthulhu-hack I use)