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