My favorite resolution system is actually a pair of systems. The first is the roll-under percentile system as done by the Design Mechanism's MYTHRAS rpg. You add to characteristics scores from 1 to 20 together to get your base skill before training. You then roll under to succeed. However, there are varying degrees of Success. Rolling under your skill is a Success but we break down the level of Success as follows. Please note that I "homebrewed" the Degrees Of Success to avoid doing math during the game. The levels of Success are...

Simple = Roll under modified skill
Outstanding Success = Roll under 1/2 Skill (this doesn't exist in RAW)
Exceptional Success = Roll under 1/10th Skill rounding UP (This is 1/5th Skill in RAW)
Critical Success = Roll DOUBLES under Skill with 00 counting as "zero, zero" and NOT "100" (RAW is 5% of Skill value).
Fumbles = DOUBLES OVER Skill (RAW is 5% of the total over Skill Level counting down from 100).

My Other resolution system is basically a D20 roll UNDER system that mimics the percentile system above. It is a modified (by me) version of GDW's old V2.2 resolution system from TWILIGHT2000, MERC2000, and TRAVELLER: THE NEW ERA rpgs.

In that system, you combine a Skill Level rated from 1 to 10 with an Attribute Rating from 1 to 10. This combination of SKILL and ATTRIBUTE was known as an ASSET and was prerecorded with all the Skill Difficulty Levels on the PC's character sheet. So, if your PC needed to figure out what their Climbing Asset was, they would combine STR, AGL, and CON together and add that to their Climbing Skill. So if the PC above had an average of 6 for their three Attributes and a Climbing Skill of 5, that PC would succeed on an AVERAGE Task on a roll of 11 or less.
Tasks would be assigned a DIFFICULTY LEVEL and these were taken from TRAVELLER but then modified. The DIFFICULTY LEVELS were...

VERY EASY Task = (3 X Skill Asset) so a typically Average PC (Asset 10) would succeed on a roll of 30 or less (a 20 always fails).
EASY Task = (2 X Skill Asset) so a typical PC would succeed on a roll of 20 or less.
ROUTINE Task = (1.5 X Skill Asset) so a typical PC would succeed on a roll of 15 or less (75% chance).
AVERAGE Task = (Skill Asset) so a typical PC would succeed on a roll of 10 or less (50% chance).
FAIRLY DIFFICULT Task = (0.75 X Skill Asset) so a typical PC would succeed on a roll of 8 or less (40% chance).
DIFFICULT Task = (0.5 X Skill Asset) so a typical PC would succeed on a roll of 5 or less (25% chance).
FORMIDABLE Task = (0.25 X Skill Asset) so a typical PC would succeed on a roll of 3 or less (15% chance).
IMPOSSIBLE Task = (0.1 X Skill Asset) so a typical PC would succeed on a roll of 1 or less (5% chance).

Modifiers would be expressed in TWO ways. The first way is as a numerical Bonus or Penalty of from 1 to 3, which would SUBTRACT FROM (for a bonus) or ADD TO (for a penalty) the PC's DIE ROLL for the Task Check. This was done both to reduce any math needed and to allow bonuses to have a larger effect on harder skill tasks. A bonus of 1 on a VERY EASY task is probably not going to make a difference to the PC... BUT, on an IMPOSSIBLE Task, it would DOUBLE your chances of success!

Optionally, for larger changes, a DIFFICULTY SHIFT could be used. You bump the chances from AVERAGE to ROUTINE or from DIFFICULT to FORMIDABLE.

In addition, you still have Tiers of Success. On a roll that is equal to or less than your base Skill Rating (that 1 to 10 number) for an AVERAGE Task, you achieve an Outstanding Success (a roll of 5 for the PC described above). IF you roll under HALF of your Skill Rating (a 2 for the PC described above), you achieve an Exceptional Success. This does get modified by the Task's Difficulty Levels though, so a DIFFICULT Task would need a 3 or less for Outstanding Success and only a 1 would net an Exceptional Success.

Finally, there was a version of ADVANTAGE & DISADVANTAGE in Twilight2013 that we adapted to our version of the game. Unskilled Task tests required you to roll TWO D20s and take the HIGHER SCORE (because this game is a roll-under system). Having an "Expert System" or other guidance gave you the ability to roll TWO D20s and take the LOWER ROLL (to represent the assistance on the Task).

The system was fast in play and allowed you to tailor tasks to fit any situation.