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