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