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