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