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