There is no definitive list of Skills. Just about any Skill can be chosen, as long as it is realistic for the Character.
Skill levels are as follows:
Mutants can have a Skill level no higher than 2 on any skill. This is due to their natural dependency on their Mutations. Since they have had their Mutations for the entirety of their lives, they have the natural tendancy to utilize a Mutation to achieve success in attempting any "chore". Therefore, the tendency is to not fully learn any skill. The Character will get frustrated after a few attemps, giving up on the learning. They will believe that they can perform the action (whatever it is) as well as anyone else, regardless of evidence to the contrary.

Skills gain bonuses or are learned only by doing. Every 15 successful tries increases a Skill level by 1. An unknown Skill becomes known on 25 tries (whether successful or not). No more than 2 tries per day can be counted for this.
Not all Skills can be self-taught. You can teach yourself only Skills that don't require very much aid. For example, you cannot teach yourself how to pilot a plane, but you might teach yourself how to drive a car. Which Skills can and cannot be self-taught will be up to your GM.
Class Skills should be chosen by your GM. There should not be a limit in number, just as long as they are closely related to the class.
The general rule for unknown Skill penalties is +10 to roll if character has a Skill related to the one attempted. +15 if not.
Here are some Skills and the appropriate Stats used to determine the proper Value:
Survival (INT/WILL) Perceive (INT/WILL) Deduction (INT)
Animal Handling (INT/WILL) Determine Direction (INT) Animal Use (INT)
Climb (STR/DEX) Swim (STR/CON) Persuade (WILL)
Barter (WILL/INT) Repair (INT/DEX) Stalk (WILL/DEX)
Pick Pocket (DEX) Use Plant (INT) Pick Lock (INT/DEX)
Disguise (INT Hide (INT/DEX) Interrogate (WILL)
Navigation (INT/WILL] Tracking (INT) Know Plant (INT)
Computer Programming (INT/WILL) First Aid 1 (INT/WILL) First Aid 2 (INT/WILL)
Pilot Auto (DEX/INT) Throwing(STR/INT/DEX) Pilot Motorcycle (DEX/INT) Computer Operation (INT)
Below are some suggested Class Skills: