I used to ride my bicycle to saxophone lessons. I have a hard back pack case for my saxophone. I would have been a lot more nervous with a gig bag. I bought my case from woodwind and brass. www.wwbw.com
They at least sell a backback cover for a hard guitar case:
http://www.wwbw.com/Summit-Guitar-Pa...-i101960.music
I didn't look at all their products. There might be a hard guitar case in backpack form. The only problem for me was a sweaty back. So a trailer or a "Sports Utility Bike" would have been an advantage in that area.