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