There are 49 Trophies that can be collected.

Bronze

Just a Scratch    
Use pit lane to repair your vehicle 50 times     

Not On My Watch    
Wreck a car in pit lane before the repair     

The Hard Way    
Finish 5 races without using pit lane     

Just That Good    
Finish 3 races without wrecking or using pit lane     

I Meant To Do That    
Drive over 15 oil slicks     

Watch Your Step    
Bump another car into an oil slick     

Didn’t See You There    
Bump another car into a hazard     

Turbo Speed!    
Complete 50 Slingshot boosts     

It Still Counts!    
Wreck your car across the finish line     

Winner!    
Place 1st on the Daytona Oval     

Show-off    
Perform a donut and go on to win the race     

2 Birds, 1 Stone    
Make 2 Rivals at the same time     

Showboater!    
Drift around 75 turns     

High Score    
Score higher than 175,000 in a single race     

Sun Was In My Eyes    
Wreck your car 15 times     

Look Mom!    
Drive over 35 ramp jumps     

Stuntman    
Perform 5 barrel rolls in mid-air     

Rocketman    
Slingshot into 25 ramp jumps     

Beat The Clock!    
Earn a trophy on any time trial track     

Stylin’    
Customize any car with an unlocked paint job    

Eat My Dust    
Place 1st in any race against your friend     

Good Effort!    
Finish a quick race season in any place but 1st     

The Real Deal    
Place 1st in a quick race season     

Friendly Rivalry    
Wreck your friend making him/her a rival     

Tough Guy    
Make 75 rivals     

Well Played    
Score higher than your friend in a quick race season     

Revenge!    
Score a payback     

That’ll Learn Ya!    
Score 10 paybacks     

Light Touch    
Score 125 slams     

Unnecessary Roughness    
Score 250 slams     

Beatdown!    
Make 3 rivals in a single race     

Thrashing!    
Make 5 rivals in a single race     

Scenic Route    
Drive over oil, dirt, sand, gravel, water, and grass     

Hot Shot    
Place 1st in any race     

Learning The Ropes    
Complete the Rookie tier in championship mode     

Showing Promise    
Complete the Professional tier in championship mode     

World Traveler    
Unlock all race tracks     

Drive By Healing    
Boost through pit lane restore     

Piece Of Cake    
Complete 10 challenges     

Over Achiever    
Complete 5 challenges in a single race

Silver

Domination!    
Score higher than 225,000 in a single race     

Time To Spare    
Earn a gold trophy on any time trial track     

Gotta Have ‘em All    
Unlock all vehicle paint jobs     

Lightning Reflexes    
Drive in boost mode for 12 seconds or more in a race     

So Close    
Earn silver trophies on all time trial tracks

Gold

That All You Got?    
Complete the legend tier in championship mode     

In The Groove    
Avoid walls and track barriers to score 2 perfect laps in one race     

Perfectionist    
Earn gold trophies on all time trial tracks

Platinum

Completionist    
Unlock all other trophies

http://www.ps3trophies.org/game/nascar-unleashed/trophies/