There are 31 trophies that can be collected.

Bronze

Boot camp
Qualified for Boot camp.

Judges' house
Qualified for the Judges' house.

On air
Qualified for the Live shows.

Focused
Rehearse 1 song in same career.

Serious
Rehearse 8 songs in the same career.

Professional
Rehearse 15 songs in the same career.

Viewers' favourite
Finish a Live show in first place.

Narrow escape
Escape a final showdown.

Brilliant
Answer quiz correctly 3 times in the same career.

Omniscient
Answer quiz correctly 8 times in the same career.

Solo
Perform 1 song in Solo mode.

Duo
Perform 1 song in Duet mode.

Battle
Perform 1 song in Battle mode.

Competition
Complete 1 Competition.

Bronze disc
Get a minimum of 6 000 points.

Silver

Star
Finish 3 Live shows in first place during the same career.

Real star
Win the competition in Star mode.

Piece of cake!
Win the contest without ever being in the final showdown.

Guardian angel
Win the contest after being in the final showdown at each stage.

Jack-of-all-trades
Perform all the songs in Solo mode.

Harmony
Perform all the songs in Duet mode.

Silver disc
Get a minimum of 10 000 points.

Addict
Sing at least 2 hours.

Keen music lover
Sing 50 songs.

Fashion victim
Wear all customisation items.

Gold

Superstar
Win the competition in Superstar mode.

Gold disc
Get a minimum score of 15 000 points.

Crazy night
Sing at least 3 hours.

Warm-up artist
Sing 100 songs.

Grand slam
Unlock all game tracks and customisation items.

Platinum

The X Factor
Complete all the other trophies.

http://www.ps3trophies.org/game/the-x-factor/trophies/