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