There are 13 Trophies that can be collected.

Bronze

Battleship
Destroy five asteroids at once.

Abstinence
Complete a stage without collecting a Power-Up.

Hyperspeed
Finish a stage in less than 60 seconds.

Jackpot
Score 1,500,000 points in a single game.

Survivor
Survive 3 stages without dying once.

Field Medicine
Collect enough Score Orbs in a single run to get two extra lives at once.

Rock'n'Roll
Destroy 50 big obstacles in a single game.

Hard to kill
Score exceeds 1,000,000 points (Survival).

The will to live
Score exceeds 2,500,000 points (Survival).

Silver

Triumph
Complete the single player campaign.

Allied Victory
Complete the cooperative campaign

Untouchable
Complete a whole sector without dying once.

Gold

High Score Hunter
Score exceeds 5,000,000 points (Campaign).

http://www.ps3trophies.com/forums/playstation-trophy-news/7434-ion-assault-hd-trophy-list.html