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 an alternate avatar.
Found 14 tickets.
Found the hidden profile screen.
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.
Found 7 tickets.
Unlocked Pony Wings.
Destroyed the first Core File.
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 all 24 tickets.
Found Missingno.