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