Does anyone know what software josh writes his sheets with?
From josh's anime sheet music

I haven't seen a software that produces that kind of result.

Just judging from the formatting, I'm guessing he's using Sibelius. However, I'm not sure what you mean by 'produces that kind of result'. If you mean how his transcriptions always turn out so well, that's completely due to his musical abilities and not because of any program =)

I also agree it looks like Sibelius. Engraving in notation software isn't so powerful, but given the time any notation/engraving application worth its salt can produce similar results to what he does. It's just a question of how determined you are on the presentation.