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