There are 33 Trophies that can be collected.

BRONZE

Finished Jeff Miller
Finished Jeff Miller.

Finished Chung Kim
Finished Chung Kim.

Finished Hans Stahl
Finished Hans Stahl.

Hotshot
Destroyed 10 enemies.

Riot
Made 100 vehicles explode.

Roadhog
Collided 100 times with traffic.

Clean Lap
Drove a lap without causing any damage.

Long Jump
Jumped 100 m.

Wrong Direction
Driven 200 m backwards.

Car-Scrap Bonus
Totalled your car.

No Speed-Limit
Driven 280 km/h for 15 seconds.

Untouchable
Used the shield 10 times.

Didn't hurt so much
Used the repair kit 10 times.

No Passage
Used the road block 10 times.

Unlimited Health
Played a deathmatch to the end without dying.

SILVER

Finished Terry McKennan
Finished Terry McKennan.

Finished Gerome Cassel
Finished Gerome Cassel.

Finished Gustavo el Loco
Finished Gustavo el Loco.

Finished Jasper Hounaar
Finished Jasper Hounaar.

Garage Owner
Unlocked every car.

Track Owner
Unlocked every track.

Clean Race
Finished one race without causing any damage.

Got Oil?
Used the oil thrower on 10 opponents.

Watch Your Wheels
Used the spike strip on 10 opponents.

Power Outage
Used the emp on 10 opponents.

Hooked
Used the grappling hook on 10 opponents.

First Online Victory
Won an online event in any mode.

Godmode
Won a deathmatch without dying.

GOLD

Finished Mario Gandetti
Finished Mario Gandetti.

Test Driver
Driven every car in the game at least once.

Track Tester
Driven a race on every track in the game.

Pilgrim
Driven 1000 km in the game.

PLATINUM

100% Clear
This trophy will unlock once you have earned all other trophies.

http://www.ps3trophies.org/game/crash-time-5-undercover/trophies/