There are 38 Trophies that can be collected.

BRONZE

First Dance
Complete your First Song

3 Star Performance
Get 3 Stars on a Song

Highway to the Stars
Get 3 Stars on 5 Songs

5 Star Performance
Get 5 Stars on a Song

Rising Stars
Get 5 Stars on 5 Songs

Golden!
Perform all Gold Moves in a Song

Solid Gold Baby!
Perform all Gold Moves in 5 Songs

Looking Good
Perform only Good Moves or better in a Song

Perfect Finish
Get a Perfect on the Last Move of a Song

On a Quest
Complete 5 Dance Quests

Winning!
Win a Battle

Wheel…of…Gifts!
Unlock an Item from the Wheel of Gifts

Just Sweatin'
Complete a Sweat Class

I Work Out
Complete a full 10 Minute Sweat Class

Getting Warmed Up
Burn 500 Calories in Sweat Mode

Stuck on Replay
Dance to the Same song 3 times in a row

Baby is in the Corner
Dance to (I've Had) The Time of my Life Alone

Dancing Friends!
Dance to We No Speak Americano with 3 others

Let's do the Timewarp Again
Dance to Time Warp twice in a row

Rickroll'd
Win a Battle as the Never Gonna Give You Up Coach

SILVER

You got the Moves!
Perform only Good Move or better in 5 Songs

Streaker!
Get 10 Perfect Moves in a row in any Song

Reflection of Perfection
Complete a Song with 80% or more Perfect Moves

There's no I in Dance
In any Dance Crew Song, All four Players Get 3 Stars or better (Four Coaches)

Dynamic Duo
Finish a Duet with 2 players getting 5 Stars on both Choreographies

Mixing it Up
Get 5 Stars on any Mash-Up

Superstar!
Get 3 Stars on All Songs (No Alternate, Mash-Ups or Battles)

Epic Winning!
Win each Battle Once

Bring it on!
Complete a full 25 Minute Sweat Class

Feel the Burn
Burn 1000 Calories in Sweat Mode

The Complete Package
Complete at least one Class from each of the 5 Sweat Programs

Blind Ambition
Get 5 Stars on a Song with Pictograms Disabled

New Sheriff in Town
Get 5 Stars on the Extreme Version of Wild Wild West

GOLD

In-Sync
In any Dance Crew Song, All four Players Get 5 Stars (Four Coaches)

Megastar!
Get 5 Stars on All Songs (No Alternate, Mash-Ups or Battles)

Spartan
Complete a full 45 Minute Sweat Class

Burn Baby Burn!
Burn 2000 Calories in Sweat Mode

PLATINUM

Just Dance Master
Obtain All Other Trophies

http://www.ps3trophies.com/forums/playstation-trophy-news/23501-just-dance-4-trophy-list.html