Here is the full list of all 40 Persona 4: Dancing All Night trophies - there are 21 bronze, 15 silver, 3 gold and 1 platinum.
Obtained all trophies
Obtained all accessories
Bought all costumes
Used a total of P$500,000 in Shopping
Triggered all dances/Bond Fevers with Nanako
Seen all Bond Fevers between Yu and Rise
Seen all Bond Fevers between Yosuke and Teddie
Seen all Bond Fevers between Naoto and Kanji
Bought half of all costumes
Unlocked highest difficulty (ALL NIGHT)
Bought all items in Shopping
50%+ Perfect clear on HARD track or higher
Danced using 2 or more items to raise difficulty
Cleared a track with each character in changed costumes
Watched all events, including pre-story
Completed the Database
Cleared all tracks in Free Dance
Margaret joins the dance
Got through to Mikuratana-no-Kami
Rescued Kyoka Ochimizu
Seen all Bond Fevers between Yukiko and Chie
Completed Chapter 4
Rescued Nozomi Nakahara
Rescued Sumomo Ujima
Watched replay data from Score
Rescued Tomoe Sayama
Triggered 30 types of Bond Fevers
Rescued Tamami Uesugi
Performed a full combo on a track
Obtained top rank (King Crazy) on a track
Completed Chapter 1
A performance ended prematurely
Beat your own high score
Cleared a track with no MISSes
Equipped accessory and began dancing
Changed costume and began dancing
Bought a costume for the first time
Paired Rise and Kanami in Free Dance
Triggered a Bond Fever at least once
Cleared 1 track in Free Dance