I've made a number of changes to the skill system for our campaign. I'm not sure this is what you're looking for, since our goal is to not only make the skill system realistic and make sense, but also to not make it a "game" within the game. I'm not interested in complex mechanics and skill challenges, I'm interested in rules that help adjudicate the action, but otherwise stay out of the way.
To that measure, I'm interested in a rule set that tells me what you can do, what you are capable of doing, and what you can't do. I'm also interested in minimizing repetitive dice rolls.
It starts with reigning in the bonuses. I'm not a fan of expertise doubling your proficiency modifier, for example. I also adjust most of the DCs up by 5 points from what's been published.
We have three levels of proficiency that is consistent across all types of proficiency (skills, attacks, saves, etc.), although there are a number of things (weapons for example) that expertise is currently not available. I'm still considering what the impact of that would be.
Non-proficient: +/- ability modifier with disadvantage
Proficient: +proficiency modifier +/- ability modifier
Expertise: +proficiency modifier +/- ability modifier with advantage
In other words, expertise increases your chance of success (and effectively raises your base by 5)
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I make extensive use of passive scores. I think this is one of the best parts of the 5e rules, just not fully developed.
Your passive score is your base. This is a measure of your actual skill. Bear in mind that advantage/disadvantage often applies and alters the base.
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Your capabilities are 20+your modifiers. Anything higher than this you cannot do. Note that advantage doesn't raise your actual ability, so in the case of expertise, your capability is the same as proficient. You just have a better chance of success.
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You only need to make a die roll if it's harder than your passive score, but less than your capability, and there is a consequence for failure. We use degrees of success/failure where appropriate. Usually measured at 5 or more, 10 or more, etc.
For example, if you have a modifier of +7, and you are trying to pick a lock with a DC of 25, but there is no pressing danger, then it's just a matter of time. I'd narrate the scene explaining that it takes a few moments to complete the task.
However, if the party is sneaking through a keep, and hear a group of guards coming down the hallway, and you need to pick the same lock, a roll is needed. It's within your capability, so you know you can do it. You roll, but only score a 21. In this case, a failure by less than 5, I'd rule it's just going to take some time, in this case 4 rounds ((DC - your roll; although you don't know that). So while your lookouts are keeping an eye out for the guards, and the fighter is getting impatient and is ready to kick in the door, you continue to attempt to pick the lock.
If you fail by 5 or more, then perhaps your lockpick gets stuck, you may need to make a second check to work it loose, and then it will still take the amount of time equal to DC - your roll.
If you fail by 10 or more, then your pick might break, or your drop it and it slides under the door, etc.
In other words, in the majority of the cases, the numbers plus a single, or at most a second roll, is everything you need to adjudicate the situation. Which means you can spend more time focusing on the action within the game, and less on the mechanics.
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I don't have tables or many specifics for degrees of success/failure because it's entirely situational. However, it doesn't have to be something that is specifically tied to the skill at hand. It's more based on the scenario.
In the example above, the PCs are attempting to secretly get through a door, so the guard patrol won't know they were there. This is a common trope in books, movies, TV shows, etc. While the skill that is used might be picking the lock, it's really the scenario as a whole that matters. For example, one possibility on a failed save is that you succeed in picking the lock, and get through the door and lock it behind you, but leave some subtle clues that you were there.
You could restrict yourself to something like your actions left scrape-marks on the lock that the guards might notice. We, on the other hand, are fine with expanding the options to something more than that. Perhaps your cloak gets shut in the door, with a part visible in the hallway. Or as you hurry to get inside and close the door, you drop your lock picks which rattle across the floor. There could be a great many possibilities.
A lot of players don't like this approach. They feel that it should only relate to the skill check at hand. The way I look at the scenario is that I only call for a check for the most difficult part of the goal. That is, of all the things that the party needs to do to get out of the hall quietly with their chosen approach is to pick the lock. So from that standpoint, I think it makes sense that failure can be more encompassing than just "you either pick the lock or you don't." I think it also makes for a more interesting narrative.
But the other aspect that we do which is probably different than most D&D games, is that if there is a failure, the players often decide what the consequences are. We use this approach for critical failures, skill checks, and even death. It may not be every time, but it's a lot of the time. We'll use guidelines (like it takes longer to do it), and we take inspiration from our favorite movies, TV, books, etc. For example, we model critical misses on things like Pirates of the Carribean, Princess Bride, and Game of Thrones fights among others.
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Again, as I said, a big part of our goal is to spend more time focusing on the action in the game than the mechanics. But we also like a robust mechanic that helps us adjudicate consistently and appropriately. That's the basics of the rules, but I'm happy to share them if you think they'll help.