Here is the full list of all 40 Persona 4: Dancing All Night (EU) trophies - there are 21 bronze, 15 silver, 3 gold and 1 platinum.
Obtained all trophies
Obtained all accessories
Cleared a track with each character in changed costumes
Seen all Bond Fevers between Yosuke and Teddie
50%+ Perfect clear on HARD track or higher
Seen all Bond Fevers between Naoto and Kanji
Seen all Bond Fevers between Yu and Rise
Triggered all dances/Bond Fevers with Nanako
Bought all costumes
Used a total of P$500,000 in Shopping
Bought half of all costumes
Unlocked highest difficulty (ALL NIGHT)
Bought all items in Shopping
Obtained top rank (King Crazy) on a track
Seen all Bond Fevers between Yukiko and Chie
Cleared all tracks in Free Dance
Completed the Database
Watched all events, including pre-story
Margaret joins the dance
Got through to Mikuratana-no-Kami
Performed a full combo on a track
Cleared a track with no MISSes
Rescued Kyoka Ochimizu
Triggered 30 types of Bond Fevers
Danced using 2 or more items to raise difficulty
Equipped accessory and began dancing
Completed Chapter 4
Rescued Nozomi Nakahara
Rescued Sumomo Ujima
Rescued Tomoe Sayama
Changed costume and began dancing
Bought a costume for the first time
Paired Rise and Kanami in Free Dance
Rescued Tamami Uesugi
A performance ended prematurely
Watched replay data from Score
Beat your own high score
Cleared 1 track in Free Dance
Triggered a Bond Fever at least once
Completed Chapter 1