I don't know about the rubber shoes, but I think that if we ask dog owners to clean up after their pets, we should also ask horse owners to clean up after their steeds.

The main trail damage I have observed due to horses comes when horseback riders ride on wet trails, leaving deep divots in the mud, which later dries and hardens, preserving the divots for a long time. I don't think rubber shoes would help with this, but I don't really know.

Horse poop seems to me to be as unsanitary as dog poop, and a lot harder to get around without stepping in it or riding through it. I think it also potentially puts non-native plants into the environment, depending on the source of the feed the horses are getting. So if the justification for asking dog owners to clean up their dog's deposits is sanitation and damage to the environment, as I have heard, then it seems the same justification warrants asking horse owners to clean up thier horses' deposits.

Personally, I think a bigger issue is safety. When I encounter a rider on a well-trained horse on the trails, I am not worried about my own safety. However, I have met poorly-trained horses on the trail, and I think they pose enough risk to other trail users that I am sometimes worried about my own safety when they are nearby. I have just enough experience with horses to be scared of what they can do, and to know that some horses are much better trained than others. I would be in favor of some kind of safety testing for horses and their riders to be sure that poorly-trained horses and inexperienced riders do not put other trail users at too great a risk.

Regardless of the poop and trail damage.  Horse owners and riders have extremely limited amount of public land that they have access to.  We are lucky to have some access at all to trails.  Yes we all need to be concious of not riding after rain.  I see as many bike wheel tracks along trails as i do horse shoe prints.  And with regards to the poop, after one rain shower the stuff is all but gone and its in the woods or fields.  Not in city park or your front lawn.  As you were saying about the less experienced riders, well hey we all cant be perfect.  As long as both bikers, riders, and runners/walkers know to be respectful of each other, especially when coming in close proximety then problems should be to a minimum.  Just remember its normally easier for you to move over/off the trail then for a horse to stay calm as you wiz bye on your bike. 

Horses enjoy historical rights of passage on some roads, let alone trails.  Check your local municipality's bylaws to see if yours still gives horses first rights of passage, over motorized vehicles even.  Horse shoes, as someone else pointed out, are for the maintenance of the hoof, so requiring rubber shoes just would not work for many.  But have you not noticed that some horses are not shod at all?  Depends on terrain and the horse and the ride.  While I agree that dog owners should clean up they are already on the ground.  For those meeting inexperienced horses and riders the last thing you want is for them to be dismounting.  Horses under riders are more in control.  It won't always be possible to tether the horse somewhere while shovelling and it is absolutely ridiculous to consider making riders carry shovels anyway.  Tail bags are equally ridiculous and apt to cause more problems with spooking than they solve with containing poop.  Horses should be exempt from silly rules on trail clean-up if for no other reason than that it could be more dangerous for all trail users to have shovels flopping and dismounted riders or tail bags causing spooking.