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