87:2 A horse can go out on Shabbos wearing a harness or a bridle but not both of them. (It is permitted to use both if one is insufficient for the particular animal in question - Mishnah Brurah 305:10.) One may wind the straps of the harness around the horse's neck and go out like that but they must be wound loosely enough so that one can quickly put his hand in-between the straps and horse's neck so that he can draw the horse if it should try to pull away. One may move the harness and put it on the horse on Shabbos so long as he doesn't lean on it because we are not permitted to lean on animals on Shabbos. However, a donkey may not go out wearing an iron bridle because this is considered an extra measure, and therefore a burden. An ox and a cow that don't require guarding may not go out with a rope around their necks unless they are accustomed to run away on a regular basis. If one ties a rope in a horse's mouth, this is considered a burden and forbidden because it will not guard the horse should the rope slip. A rope in the mouth is not the same as a harness that is attached to the horse's head.
87:3 Neither a horse nor any other animal may go out with a saddle blanket, except for a donkey because its nature is to get cold. If the saddle blanket was attached before Shabbos, the horse may go out with it on Shabbos because it's like a garment. If the blanket is not tied on, the horse may not go out with it. This is a rabbinic safeguard against the possibility of the blanket falling off and the horse's owner coming to carry it. It's not possible to tie the blanket on Shabbos because this would require leaning on the animal. When a donkey is in a courtyard, one may put a saddle blanket on it because of the cold so long as the donkey does not go out wearing it; one may not put the blanket on the horse at all (because of unnecessary exertion - MB 305:32) unless it's actually cold enough to affect a horse. Similarly, during the summer when there lots of flies that will cause discomfort to the horse, one may put a saddle blanket on it so long as he is careful not to lean on the animal while doing so. One may not remove a saddle blanket on Shabbos, neither from a horse nor a donkey, because leaving it on causes no discomfort to the animal.