There are 40 Trophies that can be collected.

Bronze

Lost Kat    
Awakened in Auldnoir.

New Challenger    
Cleared a challenge.    

True Challenger    
Cleared EVERY challenge.    

Gold Medalist     
Gold-medaled a challenge.

Silver

Ancient Game Hunter     
Defeated the rare Nevi in Rift Planes: The Ruins.    

Burning Game Hunter    
Defeated the rare Nevi in Rift Planes: The Inferno.    

Illusory Game Hunter    
Defeated the rare Nevi in Rift Planes: The Mirage.    

Frequent Flyer    
Landed 10 gravity kicks without landing or taking damage.    

Pitching Machine    
Hurled 10 objects into enemies without missing or taking damage.    

Gravitational Anomaly    
Defeated 8 enemies with one special attack.

Silver Lining     
Earned a silver or better in every challenge.

Going Underground    
Discovered every manhole.    

Lost in Time and Space     
Heard the mysterious couple’s complete story.    

Gem Collector    
Collected a total of 20,000 precious gems.

Gold

All That Glitters     
Gold-medaled EVERY challenge.

Gem Aficionado    
Collected a total of 40,000 precious gems.    

Top Cat    
Raised Kat’s reputation to 'Top Cat.'

Platinum

It’s all Relative    
Collected every trophy. Congratulations!

22 Secret

http://www.ps3trophies.org/game/gravity-rush/trophies/