There are 51 Trophies that can be collected.

Bronze

Into the DiRT    
Complete your first DiRT Tour race

Driving School    
Complete the tutorials in the Gymkhana Academy

Kick Off the Training Wheels    
Achieve a podium finish in the Gymkhana Academy

Reputation Boost    
Complete 5 bonus Race Objectives

The Extra Mile    
Complete 25 bonus Race Objectives

Crash Proof    
Use a flashback in the DiRT Tour after receiving terminal damage and then go on to win the race

Call me Ace    
Ace all trick types in a Gymkhana Championship event

Today's Forecast is…Victory!    
Win a race in all weather conditions

Assistance is Futile    
Win a DiRT Tour race without using any driver assists

Pace Setter    
Complete a Time Trial race in a vehicle from each class of Rally

Air Miles    
Win a race in every location

Self Preservation Society    
Win a DiRT Tour race in the Mini Cooper without receiving any damage

The Real Thing    
Win a Rally race in the DiRT Tour using the interior camera and the HUD switched off

California Dreams    
Achieve a podium finish in the X Games Tournament

Sub Zero Hero    
Achieve a podium finish in the Winter X Games Tournament

Rally Evolution    
Win a DiRT Tour race in a Rally car from each decade

Rising Talent    
Reach Driver Rep Level 10

Shake and Bake    
Reach Driver Rep Level 20

World Renowned    
Achieve first place finishes and pass the DC Challenge in a World Tour event

Cool Running    
Beat the Bobsleigh in the Norway Speed Run DC Challenge

No-bot Wars    
Smash every robot in a Smash Attack DC Challenge

Perfect Sprint    
Complete a clean run in a Gymkhana Sprint DC Challenge

Showcase Drifter    
Score 25,000 drift points in a Drift Showcase DC Challenge

DC Silver    
Achieve Silver medals in all of the DC Challenges

Hooning Around    
Complete 25% of the Battersea Compound Missions

Donut Addict    
Complete 50% of the Battersea Compound Missions

Burnt Rubber    
Complete 75% of the Battersea Compound Missions

Road Trip    
Complete a Joyride online with at least 3 other players

The Professional    
Complete your first Pro Tour race online

Taking the Trophy    
Win 10 online games (Pro Tour or Jam Session)

The Road Ahead    
Complete 25 online games (Pro Tour or Jam Session)

Honorable Driver    
Complete an online multiplayer race with a ‘Cautious’ rating

Steer Hunter    
Complete a game of Invasion without any negative points (Pro Tour)

Can't Touch This!    
Remain uninfected in a round of Outbreak (Pro Tour or Jam Session)

Cheeze It!    
Be the mouse in a Cat n Mouse game where your team has gone on to win the game (Jam Session)

Flag Stealer    
Steal the flag from the opposing team 5 times in a game of Transporter (Pro Tour)

Join the Party    
Win a game in each of the multiplayer party modes (Pro Tour or Jam Session)

Super Star    
Earn enough fans to achieve the Superstar title

Silver

Teacher's Pet    
Achieve a Platinum medal in all of the Gymkhana Academy tutorials

Driven    
Win DiRT Tour races using vehicles from each discipline

Eat my DiRT!    
Reach Driver Rep Level 30

Gymkhana Aficionado    
Complete all Gymkhana Championships

SuperSeries Champion    
Win the DC SuperSeries Championship

Gym-Carnage    
Score 500,000 Points in a Gymkhana Event

DC Challenger    
Complete all of the DC Challenges

DC Gold    
Achieve Gold medals in all of the DC Challenges

Platinum Performance    
Achieve Platinum medals in all of the DC Challenges

King of the Road    
Win an online race in Hardcore Mode

Gold

From DiRT to Glory    
Achieve first place finishes in all DiRT Tour events

Battered Battersea    
Complete 100% of the Battersea Compound Missions

Platinum

Platinum Trophy    
Achieve all the trophies

http://www.ps3trophies.org/game/dirt-3/trophies/

@ freezebug2 thanks for the info mate changed source.