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