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.
Unlocked Pony Lasers.
Found 3 tickets.
Learned a secret of your past.
Crashed the game.
Found your first ticket.
Learned every secret of your past.
Found 7 tickets.
Found 14 tickets.
Found all 24 tickets.
Found the hidden profile screen.
Found Missingno.
Found an alternate avatar.