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