Here is the full list of all 40 The King of Fighters Collection: The Orochi Saga trophies - there are 16 bronze, 21 silver, 2 gold and 1 platinum.
KOF96: Defeat Goenitz with a team that includes Goenitz.
Complete all Challenges listed under 'Advanced'.
Complete all Challenges listed under 'Defense'.
Complete all Challenges listed under 'Combo'.
Complete all Challenges listed under 'Secret Characters'.
Complete all Challenges listed under 'DMs'.
Complete all Challenges listed under 'Time Trial'.
Complete all Challenges listed under 'Endurance'.
KOF98: Clear the Arcade mode without using a continue.
KOF98: Defeat Omega Rugal.
KOF97: Defeat Orochi.
KOF97: Clear the Arcade mode without using a continue.
KOF97: Defeat Orochi with a single character.
KOF97: Defeat the "real" New Faces Team.
KOF94: Defeat Rugal aboard the Blacknoah.
KOF95: Defeat Rugal. Again.
KOF95: Defeat Saisyu Kusanagi.
KOF95: As Kyo, defeat Saisyu Kusanagi on your own.
Complete all Challenges listed under 'Extra'.
KOF96: Defeat Goenitz.
KOF96: Clear the Arcade mode without using a continue.
KOF96: Defeat Chizuru Kagura.
KOF96: Defeat the Boss Team of Geese, Krauser, and Mr. Big.
Unlock all other trophies.
KOF94: Complete the tournament as Team USA without losing a match.
KOF94: Clear the Arcade mode without losing a match.
KOF95: Clear the Arcade mode without using a continue.
KOF96: Defeat Goenitz with a team of Kyo, Iori, and Kagura.
KOF97: Defeat Orochi with a team of Kyo, Iori, and Kagura and see the game's true ending.
KOF97: See all of the special team edit endings.
KOF97: Defeat Orochi and his minions with Chris leading the New Faces Team, and without getting knocked out.
Complete all Challenges listed under 'Offense'.
Complete all the Challenges.
Play through the canonical version of the Orochi Saga.
Clear all five games with the Fatal Fury Team.
Clear all five games with the Art of Fighting Team.
Clear all five games with the Psycho Soldier Team.
Clear all five games with Team Korea.
Clear all five games with the Ikari Team.
Clear all five games with the Women Fighters' Team without losing a match.