There are 51 Trophies that can be collected.

Bronze

You know the Midnight Channel?
Tuned In

Champion!
You completed Arcade Mode with one character.

Halfway There
You completed Stage 6 in Score Attack

Tres Bien!
You completed Score Attack Mode with one character

School's Out
You've aced all the tests.

The Kingpin
Yu's Story chapter has been told.

The Carnivore
Chie's Story chapter has been told.

The Captain
Yosuke's Story chapter has been told.

The Unconquerable
Yukiko's Story chapter has been told.

The Emperor
Kanji's Story chapter has been told.

The Beast
Teddie's Story chapter has been told.

The Detective
Naoto's Story chapter has been told.

The Queen
Mitsuru's Story chapter has been told.

The Boxer
Akihiko's Story chapter has been told.

The Angel
Aigis's Story chapter has been told.

The President
Labrys's Story chapter has been told.

The Elevator Attendant
Elizabeth's Story chapter has been told.

Trained in Seclusion
You spent 30 minutes in Training in one sitting. Now take it to the streets!

I Learned a Few Moves
You have completed 150 Challenges.

Amazing, Sensei!
You've earned your first online victory! You've bear-ly scratched the surface.

Veteran
You have completed 100 online matches, win or lose. No bear puns here. Congrats.

Speak with Your Fists!
You've battled against every character in online matches. You're a grizzly'd warrior now!

They Call Me…
You've created a Title for your online matches

Replay Reviewer
You watched a Replay in the Theater. Are you as good as you remember?

Mystery Food X
Mystery Food X is yours!

Wild
You have used all the characters in Arcade Mode or Online. Variety is good for you!

Stylish!
You performed 100 Auto Combos in Arcade Mode or online.

Action
You performed 100 Furious Actions in Arcade Mode or online.

Time for the All-Out Attack!
You performed 100 All-Out Attacks in Arcade Mode or online.

All-Out Attack to the MAX!
You finished an All-Out Attack with a Fatal Counter in Arcade Mode or online.

Die for Me!
You performed 30 Fatal Counters in Arcade Mode or online.

One More!
You performed 20 One More Bursts in Arcade Mode or online.

Null Physical
You performed 50 Instant Blocks in Arcade Mode or online.

Per…so……na…!
Have you burned your dread? Or reached out to the truth?

Persona Breaker
You created 10 Persona Breaks in Arcade Mode or online.

Calm Down.
You suffered a Persona Break in Arcade Mode or online. It happens to everybody…

> You Should Leave Him Be…
You were afflicted with every ailment in Arcade Mode or online. Not even Mr. Edogawa can help you.

Meat Dimension
You defeated an opponent in Arcade Mode or online before he Awakened! Now, that's a real challenge!

Denied Pets
You lost in Arcade Mode or online before you Awakened… That's sad. Here's a consolation prize.

Victory Cry
You won 30 Rounds in Arcade Mode or Online by using SP Skills or Awakened SP Skills.

BAM!
You used every character's Instant Kill move in Arcade Mode or online!

Perfect!
You had a perfect victory in Arcade Mode or online.

Silver

The End of Everything
You have seen all the Story endings.

I Mastered that One
You have completed all of one character's Challenges!

Combo Master
You have completed 300 Challenges.

Black Belt
You have reached Grade C in Ranked Matches.

Serious Gamer
You have reached Level 30 by playing online matches!

Gold

Belt Collector
You completed Arcade Mode with all characters

C'est Magnifique!
You completed Score Attack Mode with all characters.

Combo Crazy
You have completed all the Challenges! Funky Student would be proud.

Platinum

I am Thou…
You have mastered P4A.

http://www.ps3trophies.com/forums/playstation-trophy-news/19412-persona-4-arena-trophy-list.html