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