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