Here is the full list of all 47 The King of Fighters XIII trophies - there are 33 bronze, 11 silver, 2 gold and 1 platinum.
Unlock all trophies.
Clear 200 trials in Trial mode
Play 300 [ranked matches / player matches]
Complete all trials for 1 character in Trial mode
Perform 50 MAX Cancels (Arcade, Versus)
Get 50 NEO MAX Super Special Moves Finishes (Arcade, Versus)
Perform 50 HD Cancels (Arcade, Versus)
Perform 50 Super Special Moves Finishes (Arcade, Versus)
Perform 50 Super Cancels (Arcade, Versus)
Hit 100 times using NEO MAX Super Special Moves (Arcade, Versus)
Hit 100 times using EX Super Special Moves (Arcade, Versus)
Hit 100 times using EX Special Moves (Arcade, Versus)
Create 10 characters in Customize
Perform 10 Perfect Victories (Arcade, Versus)
Perform 50 Drive Cancels (Arcade, Versus)
Hit 100 times using Super Special Moves (Arcade, Versus)
Perform 300 Target Actions (Arcade)
Win 100 [ranked matches / player matches]
Get 10 consecutive wins in a ranked match
Get 5 consecutive wins in a ranked match
Get 2 consecutive wins in a ranked match
Play 100 [ranked matches / player matches]
Win 50 [ranked matches / player matches]
Play 50 [ranked matches / player matches]
Win 25 [ranked matches / player matches]
Win 5 [ranked matches / player matches]
Win your first [ranked match / player match]
Create 10 rooms
Play your first [ranked match / player match]
Defeat 35 characters in Survival mode
Clear Time Attack mode with 30 characters
Be challenged by "Saiki" and win in ARCADE
Clear ARCADE MODE without continuing
Clear Time Attack mode for the first time
Clear ARCADE MODE on VERY HARD
Be challenged by "Ash" and win in ARCADE
Register an icon, team and message in Customize
Get 10 straight wins (Arcade, Versus)
Be challenged by "Billy" and win in ARCADE
Hit 100 times using Special Moves (Arcade, Versus)
Get an SS at the victory screen
Complete STORY.
Clear STORY.
Perform 10 Target Actions (Arcade)
Get an S at the victory screen
Defeat 3 characters in Survival mode
Complete the tutorial