There are 15 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)

Ranking Mode
Play Ranking Mode in Arcade Mode (hold down P+K+D+START button during Attract Mode)

Random Mode
Hold↑and hit START button during Attract Mode in Arcade Mode

Kids Mode
Select game settings in Arcade Mode or Offline Versus while holding L1 button+R1 button to activate Kids Mode

Wall Breaker
Destroy a stage wall in Arcade Mode or Offline Versus

Armor Destroyer
Break an opponent's armor in Arcade Mode or Offline Versus

Who needs armor?
Unequip armor in Arcade Mode or Offline Versus (quickly input →←→←+P+K+D)

Mahler
Use Mahler in Arcade Mode or Offline Versus (move cursor left of Grace during character selection)

Class of '95
Select game settings in Arcade Mode while holding the □ button to enter Attract Mode

Sweet Tooth
Select game settings in Arcade Mode while holding ↑ to unlock Candy's appearance in 2nd Attract Mode

SILVER

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

GOLD

Skirt Destroyer
As Candy unequip armor each Arcade stg or break her lower armor after 10 straight wins in Offline VS

http://www.ps3trophies.org/game/fighting-vipers/trophies/