There are 12 Trophies that can be collected.

Bronze

Jam Party Rookie    
Record a performance.    

Stop Watch    
Trigger a track on-beat during a performance.    

Jam Jockey    
Play a recorded jam in Arcade Mode.       

Junior Producer    
Complete 5 jams in Arcade Mode.    

Play it Again Sam    
Trigger the same track on-beat three times in a single jam.    

Loop Master    
Trigger 20 consecutive loop tracks on-beat.

Going Bronze    
Earn 20 bronze medals in Jam Mode.

Silver

Set Master    
Trigger 4 consecutive bankset changes on-beat.    

Soloist    
Trigger 20 consecutive freeform tracks on-beat.

Going Silver    
Earn 20 silver medals in Jam Mode.

Arcade Master    
Complete all jams in Arcade Mode.

Gold

Going Gold    
Earn 20 gold medals in Jam Mode.

http://www.ps3trophies.org/game/jam-live-music-arcade/trophies/