There are 12 Trophies that can be collected.

BRONZE

Perfect Host
Host and complete 10 Split-screen races.

Bang, Bang Kaboom!
Detonate 50 exploding barrels.

Tis But A Scratch
Win a Split-screen Standard Race of minimum 5 laps without using the Pit Lane.

Cup Collector
Finish every event in Career mode in Bronze, Silver or Gold places!

Rookie of The Year
Win the N-Dura Cup.

Real Racer!
Win the Evo GT Challenge Cup.

Slick on Slicks!
Win the Protech Series.

Lookin' Good!
Complete an entire Championship without ever letting your health drop below green.

SILVER

Bang Bang Party
Complete 10 Split-screen Championships.

You Beat Us!
Beat our Best Lap Time of 00:21:451 in Time Trial on Sakura Speedway Long.

Beat The Clock
Achieve Gold awards in Time Trial on all the Track variants in each Class.

GOLD

The Right Formula
Win the Apex World Trophy

http://www.ps3trophies.org/game/bang-bang-racing/trophies/