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