There are 45 Trophies that can be collected.

Bronze

Amateur Ant Killer
Kill 100 Ants.

Welcome To The EDF
Kill 1000 Ants.

Newb
Kill 100 Spiders.

Tick Picker
Kill 100 Ticks.

Tick Tock Ya Don’t Stop
Kill 5000 Ticks.

Hector? What?
Kill 10 Small Hector.

Hector Wrecker
Kill 100 Small Hector.

The Sting’s The Thing
Kill 10 Wasps.

War Against Soaring Pests Award
Kill 100 Wasps.

It’s Bomber
Kill 10 Bomber Spiders.

Look What The Rookie Did!
Kill 5000 Enemies.

It’s A Start
Destroy 100 Gunships.

Gunship Down
Destroy 1000 Gunships.

Eyes Up!
Destroy 10 Carriers.

Campaigner
Unlock All Armors.

Good Start
Make It Through 5 Survival Mode Waves In One Session.

Surviving…
Get 1000 Kills In Survival Mode.

Social Dead Butterfly
Get 10000 Kills In Survival Mode.

Smells Like Team Spirit
Incapacitate A Teammate.

Helping Hand Award
Revive A Teammate.

Bedside Manner
Revive 250 Teammates.

Lander Commander
Get 250 Kills With The Lander Turret.

Mechs Master
Get 500 Kills With The Mech.

I Heard Them Scream
Get 500 Kills With The Tank.

Nothing Turret
Get 500 Kills With Base Turrets.

The Architect
Destroy 500 Buildings.

Demolition Derby
Destroy 500 Cars/Vehicles.

Collateral
Destroy 500 World Objects.

Easy Peasy
Beat The Game On Easy.

Lemon Squeezy
Beat The Game On Normal.

Silver

Ant Farmer
Kill 10000 Ants.

Spider Fighter
Kill 10000 Spiders.

Fleet Depleter
Destroy 10000 Gunships.

Carrier Dropper
Destroy 100 Carriers.

Nesting Instinct
Destroy 10 Anthills/Spider Nests.

Scavenger
Collect All Dropped Weapons.

All Dressed Up…
Level Up An Armor Completely.

High Roller
Purchase All Available Weapons For A Single Armor.

Hangin’ Tough
Survival 100 (total) Waves In Survival Mode.

Survive-O-Rama
Survive 500 (total) Waves In Survival Mode.

Gold

Who Ravages The Ravagers?
Kill 25000 Enemies.

Homewrecker
Destroy 100 Anthills/Spider Nests.

Lifer
Level Up All Armors Completely.

Daddy WarBugs
Purchase All Weapons For All Armors.

Platinum

Platinum Trophy
Earn all other trophies in Earth Defense Force Insect Armageddon!

5 Secret

http://www.ps3trophies.com/forums/trophy-news/60715-earth-defense-force-insect-armageddon-trophy-list.html