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