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