There are 51 Trophies that can be collected.

Bronze

Drivers Ed.    
Complete the Prologue    

Back on the Road    
Complete Chapter 1    

Eyes on the City    
Complete Chapter 2    

Against the Odds    
Complete Chapter 3    

Now You See Me…    
Complete Chapter 4    

Deep Cover    
Complete Chapter 5    

Another Perspective    
Complete Chapter 6    

The Truth    
Complete Chapter 7

Initiate    
Reach Multiplayer Level 5 (Public)

Air Rage    
Spin out 10 Vehicles with Impulse (Public)    

Connoiseur    
Swap or Spawn into 20 Unique Vehicles (Public)    

Professional Racer    
Place in the top 3 in Classic Race 10 times (Public)    

Picking Up the Pace    
Place in the top 3 in Sprint GP 10 times (Public)    

Round and Round You Go    
Place in the top 3 in Shift Race 10 times (Public)    

Carry the Team    
Score 10 times in Capture the Flag (Public)    

Pass the Torch    
Carry the Torch for 20s, 25 times in Relay Race (Public)    

Relentless Assault    
Successfully attack the base 10 times in Blitz (Public)    

Tag, You're It!    
Keep the Tag for 10s, 50 times in Tag (Public)    

Getting Away With It    
Reach 25 drop-off locations in Takedown (Public)    

I'm On Fire!    
Stay in both trails for 30s in Trailblazer (Public)    

Stay With the Pack    
Score 1000 gates in Checkpoint/Team Rush (Public)    

The Challenger    
Beat the Target Time/Score on any Challenge    

Fan Service    
Complete the 'Blast from the Past' Challenge    

Lights, Camera, Action!    
Unlock a Movie Challenge    

Cut! Print It!    
Unlock 6 Movie Challenges

Active!    
Complete an Activity    

Hyperactive!    
Complete 25 Activities

We Dare You…    
Complete a Dare    

We Double Dare You…    
Complete 40 Dares

Are You Insured?    
Buy your first Vehicle    

Car Nut    
Buy 50 Vehicles

First Time Buyer    
Buy your first Garage    

Keys to the City    
Buy all of the Garages in the City    

Tricked Out    
Buy all of the Upgrades from the Garage

This Can't Be Real    
Complete the Shift Tutorial

Ramped Up!    
Complete a Getaway Activity in a Car Transporter    

Lombard Streak    
Drive down Lombard Street, no collisions, over 20 mph (32 km/h) (Story)    (4)

Show Off    
Perform a 250m Drift or Jump (Story/Online)

Silver

Survivor    
Complete Chapter 8

Veteran    
Reach Multiplayer Level 20 (Public)

That's a Wrap!    
Unlock all 13 Movie Challenges

Radioactive!    
Complete 50 Activities

Petrolhead    
Buy all of the Vehicles from the Garage

Hey Big Spender!    
Spend 5,000,000 WP

Shift Happens    
Shift 1000 times (Story/Online)    

Due for a Service    
Drive over 1000 miles (~1610 km) (Story/Online)    

Cops & Robbers    
Complete 30 Freedrive Cop Chases

Gold

End of the Road    
Complete the Story

Master    
Reach Multiplayer Level 38 (Public)

Platinum

You ARE the Wheelman    
Collect all Trophies

http://www.ps3trophies.org/game/driver-san-francisco/trophies/