Mini-games and Modes:
Unlock the following modes and mini-games by performing the corresponding tasks:

Katamari Drive Mode - Complete the game once and see the ending
Beautiful Katamari Hard Minigame - Beat Katamari Drive Mode
Beautiful Katamari Normal Minigame - Beat Katamari Drive Mode
Katamari Forever Ending Credits - Beat the game

Music Selection:
To unlock music options for a specific level, complete that level.

Graphics Filter Selection:
To unlock graphic filters for a specific level, complete that level.

Trophies:
Big Katamari (Bronze) - Create a Katamari over 500m.
Collector (Bronze) - Complete 80% of the collection.
Completionist (Gold) - Complete 100% of the Collection.
Congratulations! (Gold) - Complete all Requests in Katamari Forever mode.
Cousin Friendly (Bronze) - Roll up 30 Cousins.
Creator (Silver) - Creat 200 katamaris.
Endurer (Silver) - Obtain 50,000 points in the punishment mini-game.
Enormous Katamari (Silver) - Create a katamari over 2,000,000km
Eternal Roller (Silver) - Complete all Requests in Eternal mode.
Fashion Conscious (Bronze) - Collect 30 Presents.
Fast Roller (Bronze) - Perform the Dash exactly 6 times before completing a Request.
First Katamari (Bronze) - Complete a request.
Hi-Scorer (Silver) - Obtain 50,000 points in the ending mini-game.
Hopper (Bronze) - Perform the Prince Hop exactly 7 times before completing a Request.
Huge Katamari (Bronze) - Creat a Katamari over 10,000km.
Katamari Destroyer (Bronze) - Destroy the katamari 50 times before completing a Request.
Katamari Fanatic (Gold) - Complete all Requests in Classic Katamari mode.
Katamari Speedster (Gold) - Complete all Requests in Katamari Drive mode.
Mini-Gamer (Bronze) - Play through all mini-games.
Movie Fan (Bronze) - Watch 5 movies in a row in Movies.
Multiplayer (Bronze) - Play through VS Battle or Co-op 30 times.
Music Lover (Bronze) - Listen to 5 songs in Music.
Photographer (Bronze) - View a photo in the Photo Album.
Platinum Katamari - Collect all trophies
Present Collector (Gold) - Collect all Presents.
Quick Turn Expert (Bronze) - Perform the Quick Turn exactly 5 times before completing a Request.
Rankings Lover (Bronze) - View the Network Rankings 5 times.
Rolling Expert (Bronze) - Roll up 7,650 objects before completing a Request.
Single Player (Bronze) - Play through 50 single player Requests in a row.
Stardust (Bronze) - Turn a katamari into stardust.
Unforgiving (Bronze) - Punish 50 times.
We Love Cousins (Gold) - Roll up all Cousins.
We Love Stardust (Silver) - Turn 50 katamaris into dust.