There are 11 Trophies that can be collected.

Bronze

Strategic Mind
Complete all levels in Puzzle Mode

Perfect Accuracy
Destroy every single tile on the game field

Brave New World
Complete the first level of the Antique Age playing for Jack

Blocker Killer
Match a line of blocker tiles

Make Them Smile
Activate the super bonus 3 or more times while playing a story level

Silver

Combo Master
Make 15 combos during a level

Last Gasp
Complete a level when there's only 8 seconds of time limit left

I'm the Bonusman
Activate 50 or more bonuses during a level

Ultimate Combo
Make the best possible combo in the game

Sleuth
Complete all hidden objects levels playing for Ilona or Jack

Gold

On Wheels
Complete 20 story levels on hard mode without losing

http://www.ps3trophies.org/game/time-machine-rogue-pilot/trophies/