There are 38 Trophies that can be collected.

Bronze

Island Raiser    
Use the Plant Golem to raise an island out of the sea.

Beginning Salvager    
Successfully salvage 10 times.     

Veteran Salvager    
Successfully salvage 20 times.     

Master Salvager    
Successfully salvage 30 times.     

Rock Hard Riddle    
Get to the bottom of why Sprout Island turned to stone.     

Monster Battler    
Defeat 100 monsters.     

Monster Masher    
Defeat 500 monsters.     

Monster Destroyer    
Defeat 1500 monsters.

Giant Battler    
Defeat 10 giant monsters.     

Giant Slayer    
Defeat 20 giant monsters.

Beginning Crafter    
Successfully craft an item.     

Veteran Crafter    
Successfully craft 300 items.

Failed Chef    
Fail at cooking one time.     

Insomniac    
Stay awake until you collapse at 5AM.

Beginning Angler    
Successfully land a fish.     

Veteran Angler    
Successfully land 300 fish.

Jumping for Joy    
Perform a double-jump 500 times.

Trophies 101    
Listen to Maerwen's explanation about trophies.

Silver

Island Savior    
Raise all of the islands out of the sea.

That Which Monsters Fear    
Defeat 3000 monsters.     

That Which Monsters Flee    
Defeat 5000 monsters.

Giant Destroyer    
Defeat 30 giant monsters.

Professional Crafter    
Successfully craft 800 items.

Most Popular    
Earn the goodwill of many residents on Fenith Island.

Professional Angler    
Successfully land 1000 fish.

Overmedicated    
Consume 100 recovery drinks.     

Germ Factory    
Catch a cold 100 times.

Gladiator     
Win a battle in the colloseum.

Gold

Master Crafter    
Successfully craft 1500 items.

Master Angler    
Successfully land 2000 fish.

Platinum

Rune Factory Master    
Earn all trophies.

7 Secret

http://www.ps3trophies.org/game/rune-factory-tides-of-destiny/trophies/