Here is the full list of all 59 Songs for a Hero - there are Definitive Edition achievements.
Find the last hidden useless spot.
Collect 9999 Coins
Survive The Asteroids.
Become The Super Hero
Solve The Pyramid Puzzle.
Kill All Enemies During The Avalanche.
Use A Box To Slide On The Cliffs
Find All Energy Container PIeces
Kill 20 Enemies During The Tornado.
Use The Minecart.
Survive The Roling Stone.
Kill 200 enemies
Finish The Epilogue
Find All Heart Container Pieces
Kill The Epilogue Sub Boss
Collect All HERO Letters
Mimic The Adventurer's Jump
Kill 100 enemies
Pet The Dog
Finish The Second World
Finish The Third World
Finish The Fourth World
Finish The Fifth World
Finish The Sixth World
Finish The Seventh World
Finish The Eighth World
Defeat The First Secret Boss
Defeat The Second Secret Boss
Defeat The Third Secret Boss
Defeat The Fourth Secret Boss
Defeat The Fifth Secret Boss
Defeat The Sixth Secret Boss
Defeat The Terrible Villain
Unlock the Flaming Sword
Unlock the Invisible Potion
Unlock the New Shield and Dive Skills
Unlock the Dash
Unlock the Double Jump
Unlock the Hookshot
Unlock the Crossbowmerang
Learn All Sword Moves
Collect All Musical Notes
Collect Half Musical Notes
Open The Frozen Chest
Finish The First World
Defeat The Sixth Secret Boss While Taking No Damage
Defeat The Fith Secret Boss While Taking No Damage
Defeat The Fourth Secret Boss While Taking No Damage
Defeat The Third Secret Boss While Taking No Damage
Defeat The Second Secret Boss While Taking No Damage
Defeat The First Secret Boss While Taking No Damage
Defeat The Seventh Boss While Taking No Damage
Defeat The Sixth Boss While Taking No Damage
Defeat The Fifth Boss While Taking No Damage
Grab All The Minecart Coins
Defeat The Fourth Boss While Taking No Damage
Defeat The Third Boss While Taking No Damage
Defeat The Second Boss While Taking No Damage
Defeat The First Boss While Taking No Damage