There are 12 Trophies that can be collected.

Bronze

Teamwork
Clear at least 2 lines with a piece swapped to you in Shared (additive)

Lockpick
Unlock all Variants

Giver
Pass 100 lines to opponents on any multiplayer mode (additive)

Next!
Surpass a Friend on Next Best Score

Synched
Get a Tetris within 3 seconds of a friend getting a Tetris in Team Battle

Tetrinet
Finish all 4 Multiplayer modes online

Trigger Happy
Send 12 or more lines to one opponent in under 5 secs in Battle, Timed Battle or Team Battle

Silver

Advanced
Complete all Advanced Feats

Achiever
Complete all Career Feats

Insane
In Origin clear lines 20, 10 and 1 in that order

Clear as Water
Perform a Bravo (clear your matrix)

Gold

Master
Finish all variants on level 15 and Radical

http://www.ps3trophies.org/game/tetris/trophies/