There are 12 Trophies that can be collected.

Bronze

Let's Do This Thing Right!
[ARCADE MODE] Defeat 10 or more Privates.

Stage Clear
[ARCADE MODE] Clear Stage 1.

All Clear
[RISING MODE] Clear all missions.

Decorated Soldier
[RISING MODE] Obtain all medals.

Burning Spirit
[RISING MODE] Achieve the highest rating on all stages.

You Can Never Play Too Much
[RISING MODE] Purchase all obtainable items.

The Strongest Warrior
[ARCADE MODE] Clear all missions.

Silver

Innate Guerrilla Warfare
[ARCADE MODE] Clear all missions without defeating more than 50 Privates.

Disposable Warrior
[ARCADE MODE] Stock up 20 or more extra lives.

World Peace
[ARCADE MODE] Clear all missions with friends around the world.

The Definition of a Warrior!
[ARCADE MODE] Clear all missions without picking up any items.

Gold

I'll Make Them Pay…
[ARCADE MODE] Clear all missions without dying.

http://www.ps3trophies.org/game/hard-corps-uprising/trophies/