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