There are 12 Trophies that can be collected.

BRONZE

Stage 1 Complete
Complete Stage 1 of Arcade Mode (any difficulty)

Stage 2 Complete
Complete Stage 2 of Arcade Mode (any difficulty)

Stage 3 Complete
Complete Stage 3 of Arcade Mode (any difficulty)

Stage 4 Complete
Complete Stage 4 of Arcade Mode (any difficulty)

Stage 5 Complete
Complete Stage 5 of Arcade Mode (any difficulty)

Stage 6 Complete
Complete Stage 6 of Arcade Mode (any difficulty)

Stage 7 Complete
Complete Stage 7 of Arcade Mode (any difficulty)

SILVER

Barrier Destroyer
Destroy all of an opponent's barriers while playing Arcade Mode or Offline Versus

A Small Transformation
Turn small by being attacked (ie, by Amy's↓+P, etc) in Arcade Mode or Offline Versus

Honey the Cat
Use Honey the Cat in Arcade Mode or Offline Versus (Press START button on Amy and select)

The Enemy Within
Fight a mirror match in Arcade Mode or Offline Versus

GOLD

Perfect
Win a round without receiving any damage in Arcade Mode (any difficulty)

http://www.ps3trophies.org/game/sonic-the-fighters/trophies/