My ideal system would be some sort of logarithmic/rank progression (takes more points to go from 2->3 than 1->2) and maybe some sort of skill rank.
Pendragon skill system is pretty great. It's a simple d20 roll under your skill. Progression is that if you get a 'check' for your skill (generally left up to GM to decide when that happens which is a bit of a weakness, but can be amended), you roll the skill in your Winter Phase bookkeeping and if you fail the roll, the skill goes up (or if your skill is 20 or higher, it goes up on a roll of 20. Getting skill to 20 makes fumble impossible, skills over 20 have a higher chance of critting.
To cut down on randomness, you get to train during Winter Phase and get either 1d6+1 points to distribute between skills under 15 or increase a skill of 15 or more by 1 up to 20.
It even has your background affect skills in that the expanded character creation adds cultures. The Cymrics (default culture) have Lance Expertise which basically rolls three skills into one (Lance, Spear and Great Spear). Other cultures get different bonuses: Aquitainians, for example, get their Romance and Courtesy skills rolled into one. Which is much less useful, but Cymric knights are supposed to be the best by design.
I think the fundamental problem with D&D skills is d20 vs target pass/fail isn't a very good skill system, but the designers of the game have, for the last 25 years, felt a relentless need to unify everything under the same engine. It works well enough for combat (but isn't particularly good for that, either).
d20 is just way too swingy and it never really worked well for me unless it was a roll under system. Sometimes I wonder how much would using 2d10 instead of 1d20 change things, but never enough to do even the most cursory math.
In the end, I can't remember any system that had a fantastic skill system. It's relatively easy to abstract comparative combat abilities and how difficult a task with dynamic difficulty like attacking an enemy is, but not so much specific non-combat abilities like skills.
HERO System 6E does, but that's a system that puts a lot of work onto players. The skills in it are supposed to be used in heroic play, rather than superheroic and it shows. The basic system is 3d6 roll under with 11- being the 'average' roll. You can spend 1 point on a skill to make it 8-, 2 points total to make it 11- and 3 points total the skill roll your stat roll. (Each skill is keyed to a stat and stat roll is 11- at 10 in a stat and it goes up by 1 for every 5 points in a stat).
After that you can buy skill levels for 2 points per +1 or penalty skill levels for 1 point per +1, but penalty skill levels only mitigate penalties. There's an 'Everyman skill' option that's just skills that get the 8- rank for free.
The Skills book just expands for it a lot and has a ton of examples and modifiers for the skills. For example, it's -8 to Charm when trying to seduce a completely biologically incompatible alien.
It also has four pages devoted to mutual intelligibility of pretty much every real life language that has at least a million speakers.
The Bases book expands on it by letting you purchase facilities in your base that help with Skills. Like a state-of-the-art Surgery room for Medicine rolls or a specialized library for specific Knowlegde skills.
You usually want to stack modifiers and the easiest way is to use extra time. If an action takes one turn (12 seconds), you can take an hour to get a +4 or +5. It's simple to adjudicate and easy to calculate, since you never work with big numbers. For things like picking locks, I personally allow adding the bonus after the roll as in "You're two short, so do you want to spend 30 minutes picking the lock or do you try something else?"
High-power superheroic characters are built with so many points that they often get 18- roll on skills with just their base stats, so the system kind of falls by the wayside at that power level. Superman doesn't defuse a bomb, he takes it and flies so high that it harms no one when it detonates.