Well my town doesn't have a team, but one of my teammates from my old gym that I'm still friends with is on one. They require:
Floor: Round off back handspring layout, front layout front pike or front handspring punch front front pike, 180 degree switch leap or 180 switch side, double full turn (those are the big things)
Beam: 1 flight (bh, round off), BWO BWO or cartwheel cartwheel, front layout or bhs tuck dismount, full turn, 180 leap or 150 switch leap (again, big skills)
Bars: in order to compete they really look for a kip cast handstand free hip handstand, pike on or sole circle, long hang kip cast handstand, free hip handstand giant layout flyaway OR 2 consecutive giants layout flyaway
Vault: tsuk or yurchenko or twisting front vault
I'm interested to see other replies!