There are 33 Trophies that can be collected.

Bronze

The Usual Nonsense
Level 1 finished.

Answer All Questions
Level 2 finished.

Hotshot
Destroyed 10 enemies.

Riot
Made 100 vehicles explode.

Rampage
Trashed 1000 Objects.

Roadhog
Collided 100 times with traffic.

Clean Lap
Drove a lap without causing any damage.

Long Jump
Jumped 100 m.

Free Fall
Fell 30 m.

Wrong Direction
Driven 200 m backwards.

Car-Scrap Bonus
Totaled your car.

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

First Online Victory
Won an online event (ranked or standard) in any mode.

Unlimited Health
Played a deathmatch to the end without dying.

Silver

The Syndicate
Level 3 finished.

Alone against the Evil
Level 4 finished.

Garage Owner
Unlocked every car.

Track Owner
Unlocked every track.

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

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

Peeping Tom
Installed every single camera possible.

Police Questioning
Activated every single informant possible.

Race King
30 Laps on racing track "Sprit-Raceway (R)".

Off-road King
30 Laps on off-road track "4x4 Club Circuit (R)".

Amateur
Won 10 single races.

State of Emergency
Caused €1 million damage during one race.

Clean Race
Finished one race without causing any damage.

Godmode
Won a deathmatch without dying.

Gold

By all Means
Finish your career successfully.

Vision becomes Truth
100% of the Single Player Game completed.

Pro
Won 50 single races.

Pilgrim
Driven 1000 km in the game.

Platinum

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

http://www.ps3trophies.com/forums/playstation-trophy-news/7408-crash-time-4-syndicate-trophy-list.html