There are 12 Trophies that can be collected.

Bronze

Welcome to the Faction   
Gain Your First Rank

Helping hands   
Complete Your First Local Multiplayer Game

Taste of Victory   
Win an Online Game

The Black Hole   
Kill All Other Players With The Singularity Bomb in a 4 Player Online Game

Total Carnage   
Kill 5 Opponents in a Row Without Dying in an Online Multiplayer Game

Scrap Metal   
Get a Quad Kill in the Training Missions

The Running Man   
Earn a Gold Star in a Survival Training Mission Without Using Any Weapons

Silver

The Skills to Pay the Bills   
Unlock All of the Training Missions

Bling   
Gain a Gold Badge in 5 Player Awards

Take That, Winters!   
Gain a Gold Star in All Training Missions

Going Up in the World   
Get To Rank 10

Gold

Faction Leader   
Get To Rank 20

http://www.ps3trophies.org/game/red-faction-battlegrounds/trophies/