There are 45 Trophies that can be collected.

BRONZE

Planet Star
Star a planet in Story mode

Battler
Achieve a 5 hit streak in any Story battle level

Showing Who's Boss!
Hit 50 Hoard minions in Story mode with any weapon

Road Rage
Slap another Sackboy on any level

Big Combo
Accumulate a 5x score multiplier in a Story level

Kart Mastery
Complete all the tutorials on the first planet, LittleBigPlanet

Just Getting Started
Spend 30 minutes in Create mode

Scholar
Watch all of the creation tutorials from start to finish

Interior Decoration
Create a custom pod and save it

Makeover
Change your costume and save it

Hot Wheels
Change your ride and save it

Making The Community
Build and save any level created in Create mode

Celestial Decorator
Create a personalized planet and save it

Custom Object
Create a custom object inside one of your levels

Arms Creator
Create a custom weapon in Create mode

Playing the Popular
Play any level in your Recent Activity

Snapper
Publish a photo taken from any level

Save This For Later
Add a level to your queue

Get Gathering
Invite three other players into your playgroup

Active Recommendation
Complete a level a friend has written a review for

We Have Good Taste
Complete a level in Team Picks

First Impressions
Complete a level with less than 5 playthroughs and leave a review

About Me
Completely populate your profile page

Better Than You
Beat a score posted by a friend on any community level

SILVER

Online Sampler
Complete 25 community levels

Finale Star
Star every 'finale' level in the main story

Crowd Pleaser
Publish a level and have 10 different players complete the level

Online Champ
Place 1st in 25 different online levels with other human players

Dive In!
Dive into 25 community levels and complete those levels

Creator Maker
Create three cooperative levels and three versus levels

GOLD

Complete Story Mode
Complete the main path of Story mode

Story Star
Star every level in the Story mode

Building the Imagisphere
Spend 24 active hours in Create mode

The Adventures Continue
Complete 100 community levels

PLATINUM

100% Complete
Earn all the LittleBigPlanet

10 SECRET

http://www.ps3trophies.org/game/littlebigplanet-karting/trophies/