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