There are 50 Trophies that can be collected.

Bronze

Balletic    
Complete a clean run in a Head 2 Head event    

Bi-Winning    
Complete a Splitscreen event    

Burst the Bubble    
Be in 1st place during each elimination period in an Eliminator event    

Crash-Bang-Wallop – What a Video    
View a Crashback replay    

Demolition!    
Wreck your first opponent in a Rampage event    

Don't Push Me    
Get the First Blood bonus in a Knock Out event    

Huntsman    
Win a Smash Hunter event    

Jack of All Trades    
Complete a Race Off, Rampage and Head 2 Head event    

No Turning Back    
Send a Challenge to a Friend    

Pro    
Achieve a podium in the Pro final    

Zero to Hero    
Dominate all sectors during the final lap in a round of a Domination event     

Comeback King    
Be in last place and then earn a podium in the double points period of a Rampage event in the Showdown Tour    

Showdown Tour    
Complete a Showdown Tour event in each location    

Mixed Grill    
Pull off an Insane T-Bone in Rampage    

Welcome to the Show    
Complete your first Showdown Tour event    

Gonna Need a Bigger Garage    
Own 15 vehicles    

Ego Boost    
Win a Race-Off event without boosting    

Hard Charger    
Use a full tank of boost in one go without crashing in a Race-Off event    

All Star    
Achieve a podium in the All Star final    

Champion    
Achieve a podium in the Champion final    

Team Awesome    
Win an Online Team event    

Treasure Hunter    
Collect all 100 Hidden Packages in Joyride    

Rust in Pieces    
Wreck 20 vehicles in the Lafayette Eulogy    

Beat it!    
Accept and beat a Friend's Challenge    

Bronze!    
Complete the Showdown Tour with at least a Bronze in every event    

Showdown Showroom    
Own every vehicle in the game    

Dock Worker    
Complete 50% of the Yokohama Compound Missions    

Hitman    
Achieve all hit types in Rampage    

London Calling    
Complete 50% of the Battersea Compound Missions    

Loot Carrier    
Hold the loot for 2 minutes in Smash & Grab    

Stylin' and Profilin'    
Score 300,000 points in a Trick Rush event in the Showdown Tour    

Mmmm Donuts!    
Complete the Donut King Mission in the Yokohama Construction Site area    

Money Maker    
Earn a total of $1,000,000    

Showdown Haulage Co.    
Drop the Flag off 5 times in a round of Transporter

Showdown Superstar    
Reach Fanbase Level 30 Online    

Survivor    
Survive for at least 2 minutes in a Hard Target event    

Household Name    
Reach Fanbase Level 10 Online

You've Created a Monster    
Fully upgrade a vehicle

Silver

Silver!    
Complete the Showdown Tour with at least a Silver in every event

Wrecker    
Achieve a 5x Wrecking Streak in Rampage

Checkpoint King    
Win a Speed Skirmish event    

Daredevil    
Complete 100% of the Battersea Compound Missions    
 

Hoonzilla    
Complete 100% of the Yokohama Compound Missions    

Legend    
Achieve a podium in the Legend final

Running the Show    
Dominate a Friend in Challenge mode    

Hoonigan    
Complete all Joyride Missions in both locations

Gold

Max Power    
Fully upgrade all vehicles    

Rising Star    
Reach Fanbase Level 20 Online    

GOLD!    
Complete the Showdown Tour and win Gold in every event

Platinum

Platinum Trophy    
Achieve all other Trophies

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