Here is the full list of all 42 Persona 3: Dancing in Moonlight trophies - there are 26 bronze, 12 silver, 3 gold and 1 platinum.
Obtained all trophies.
Watched every possible dance for the P3 Protagonist (Makoto Yuki).
Watched every possible dance for Aigis.
Cleared the stages where Yukari and Junpei dance together.
Registered a costume as a Favorite.
Unlocked all ???s in Collection.
Watched all Social events.
Unlocked all Custom modifiers.
Obtained all of the accessories.
Obtained all rewards from Elle-P's card search.
Cleared the stages where Fuuka and Ken dance together.
Cleared the stages where Akihiko and Mitsuru dance together.
Watched 40 different Fever Time dances.
Watched a replay.
Cleared a stage on HARD or harder with all Support modifiers enabled.
Cleared a stage with both "Note paths waver" and "Note speed fluctuates" enabled.
Obtained all the costumes.
Cleared a stage with both "GOOD ratings won't break combos" and "Game Over with combo break" enabled.
Cleared a stage with 5 Challenge and 5 Support modifiers simultaneously enabled.
Obtained all the pairs of glasses.
Explored all the guys' rooms.
Had Elle-P join the ball.
Made Elle-P's ball a success.
Cleared a stage after changing Hype Gauge difficulty in Custom.
Explored all the girls' rooms.
Started a stage on ALL NIGHT difficulty with 10 or more Challenge modifiers enabled.
Obtained half of all the accessories.
Obtained 5 rewards from Elle-P's card search.
Unlocked the highest difficulty level (ALL NIGHT).
Started a stage with a different hairstyle, two accessories, colored contacts, and a colored wig.
Explored a friend's room.
Obtained half of all the costumes.
Earned the highest possible grade on a stage (KING CRAZY).
Watched 20 Social events.
Watched a Social event.
Changed costumes and played a stage with each of the members of SEES (excluding Koromaru and Shinjiro).
Failed a stage.
Cleared a stage on HARD or harder and earned a PERFECT rating on at least half of the Notes.
Beat your own high score.
Cleared a stage.
Played a tutorial.
Obtained a costume.