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