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