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.
Found all 24 tickets.
Found Missingno.
Learned every secret of your past.
Found 14 tickets.
Learned a third secret of your past.
Found the hidden profile screen.
Escaped the arcade.
Destroyed the third Core File.
Killed Louey.
Learned another secret of your past.
Destroyed the second Core File.
Learned a secret of your past.
Found an alternate avatar.
Found 7 tickets.
Unlocked Pony Wings.
Destroyed the first Core File.
Unlocked Pony Lasers.
Found 3 tickets.
Crashed the game.
Found your first ticket.