There are 13 Trophies that can be collected.

Bronze

Disposable Heroes
Lose 10 monsters in the course of play.

Megabucks Monster Club
Cause over $850,000 worth of property damage within a level.

Monster Mash-Up
Compete in a multi-player game.

I Wonder What's for Dinner
First time monster eats.

O.C.M. (Obsessive Compulsive Monster)
Totally destroy every building in a level.

Monster Games
Run over 1000m, throw an object over 50m in a single throw, and jump over 50m in a single jump. All in one level.

Twinkle toes
Complete a race level and cause less than $50,000 worth of damage.

Inspector Monster
View every monster part in the Monster Lab.

Creature Feature
Make your first monster.

Silver

Mwuhahahahaaa
Complete all missions in every level.

Monster Modelling School
Build and use 6 different monsters in the game.

Demolition Master
Destroy 100 buildings.

Gold

Domination
Complete all missions in every level with all awards.

http://www.ps3trophies.org/game/eat-them/trophies/