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