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