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