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