There are 21 Trophies that can be collected.

BRONZE

Thorough
Earn all stars in a single mission in the Humanity Inc.campaign.

Pinnacle of Man
Complete all missions in the Humanity Inc. campaign.

Man of the Year
Reach level 20 for Human race.

Jack of all trades
Win in each basic game mode: Elimination, Capture, Domination, King of the Hill.

Deep Blue
Reach level 10 for Robot race.

Heart of iron
Win a 4 player free for all match against 3 hard AI.

Seriously?!
Destroy one million enemy population. Yes, 1 000 000.

Intergalactic
Complete 30 online multiplayer matches.

Don't panic!
Successfully defend a planet with just a few citizens on it (except for offline multiplayer).

Narrow escape
Capture a planet when you have no planets left in any mode except for offline multiplayer.

Private
Get 5 stars in the Humanity Inc.campaign.

Midshipman
Get 10 stars in the Humanity Inc.campaign.

Lieutenant
Get 20 stars in the Humanity Inc.campaign.

Commander
Get 30 stars in the Humanity Inc.campaign.

Captain
Get 40 stars in the Humanity Inc.campaign.

Vice Admiral
Get 50 stars in the Humanity Inc.campaign.

Admiral
Get 60 stars in the Humanity Inc.campaign.

Exterminator
Purge all Swarm planets before completing the mission.

Tactician
Trick Destrobos' death planet into attacking its own ally during the boss battle.

Team Player
Win an online team multiplayer match.

Half way
Complete half of the Humanity Inc.campaign.

http://www.ps3trophies.org/game/planets-under-attack/trophies/