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