I haul one horse in a two horse straight load trailer and I agree with whats been posted already. You do have to be careful but i've personally never noticed a difference as I'm very careful when hauling my babies in the back and I presume any horse person would feel the same. I also agree with letting the horse decide which position it decides is best to travel. I let my filly and all previous horses (when hauling a single horse) travel untied and have yet had a problem with it. I watched my grandpa do it and at first was scared the horse might falll down but have yet to witness one go down in the trailer. They all seem to find that position and stick to it. I like doing this especially for long trips; my filly and I recently did an 800+ mile trip in 113F this really helped her move to areas where there was good circulation of air and stay comfortable. I've also noticed they tend to go slanted and backwards which I've found out is because horses are more likely to founder when facing foreward as opposed to backwards. My final reason for letting the single traveling horse go untied is because if they were to fall in the trailer theres nothing holding them which could cause them to panic (which is why people use panic release trailer clips).
Maybe I'm off base for letting em go freestyle
but it's how we do it