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