Here is the full list of all 20 Pony Island achievements. It takes around 4-5 hours to unlock all of the achievements on Windows.
Escaped the arcade.
Destroyed the third Core File.
Killed Louey.
Learned a third secret of your past.
Destroyed the second Core File.
Learned another secret of your past.
Unlocked Pony Wings.
Destroyed the first Core File.
Found 7 tickets.
Unlocked Pony Lasers.
Learned a secret of your past.
Found 3 tickets.
Crashed the game.
Found your first ticket.
Learned every secret of your past.
Found 14 tickets.
Found all 24 tickets.
Found the hidden profile screen.
Found Missingno.
Found an alternate avatar.