There are 34 Trophies that can be collected.

Bronze

Centre Stage
Play your first Solo Dance

Dance-Off
Play your first Dance Battle

It Takes Two
Play your first Dance Partners routine

Dance Starter
Get a 5-star rating on a Beginner, normal length routine

Raise The Roof
Get at least a 'Cool Singing' rating in 5 performances

Hi-Five!
Get 5 hi-scores

Choreographer
Create a Solo Dancer routine

Dance Made For Two
Create a Dance Partners routine

Fresh New Moves
Play a Creator Routine

Get Sweaty
Perform a Dance Workout

Saving The Day
Save 20 highlights

Exclusive Footage
Upload a video or snapshot highlight to the Community Gallery

Social Networking
Share a video or snapshot highlight with Facebook or Twitter

I Like The Way You Dance
Rate a video or snapshot highlight in the Community Gallery

Party Time!
Play a 5-round Party

Beginner Class
Get a 5-star rating on every section of a Beginner routine in Dance Class

Advanced Class
Get a 5-star rating on every section of an Intermediate routine in Dance Class

Silver

Keep On Moving
Play 50 dances

Dance Champion
Get a 5-star rating on 5 Intermediate, normal length routines

Never Dropped A Beat
Achieve a 70-move streak in any routine (excludes Creator Routines)

Pitch Perfect
Get a 'Flawless Singing' rating in 10 performances

Chart Buster!
Get 40 hi-scores

Tour De Dance
Play 15 dances in one game session

Feel The Burn
Perform 10 Dance Workouts

24 Round Party People
Play 24 Party rounds in total

Learn With Friends
Get a 5-star rating on every section of a Dance Partners routine in Dance Class

Top Of The Class
Get a 5-star rating on every section of a Professional routine in Dance Class

Gold

Dance Deity
Get a 5-star rating on 10 Professional, normal length routines

Two Against The World
Get a 5-star rating on 10 Dance Partners, normal length routines (excludes Creator Routines)

Unstoppable Groove
Score 100,000 on a Professional routine

Feet Of Fire
Dance for 10 hours in total

Self Improvement
Burn at least 200 kcals a week for a month

Platinum

The Dance Never Ends…
Unlock all other trophies

1 Secret

http://www.ps3trophies.com/forums/ps3-trophy-news/2987-dancestar-party-trophy-list.html