These days one can source ready-made horse "forms" (e.g. Van Dykes) on which to mount a hide. Historically the creation of the form was one of the most difficult tasks requiring the work of a skilled naturalist.
Taxidermy has seen something of a revival in recent years, especially in the fine arts. Stuffed horses have appeared in a number of prominent art installations. For example:
Taxidermy is used to preserve horses famous as racers, performers, or military use.
Comanche, a horse that survived Custer's last stand was a celebrity in life and on his death in 1891 preserved and has long been in the possession of the University of Kansas (although his ownership is often disputed).
Le Vizir, a favorite mount of Napoleon is in somewhat poorer shape as is to be expected for an animal preserved in 1829. He is on display at the at the Musee d’Armee de l’Hotel des Invalides in Paris.
Horses may also be preserved and mounted if their body presents elements of scientific interest.
The preservation of horses opens the door to some interesting research in cases where DNA can be extracted. For example providing DNA samples for the extinct quagga, and investigation into the cause of death of Phar Lap (allegedly arsenic poisoning).
Why do so many of then look terrible?
Mounting an entire horse is a difficult job, and many who attempted it (especially in decades and centuries past) have simply not done a very good job. the science of tanning, mounting and preservation has advanced a great deal even inn the last 55 years--and older specimens often lack correct shape or are imperfectly preserved.
However many other factors contribute to poor appearance, especially in antique specimens. With time hides tend to contract and become poorly fitted to their frame. While wear and tear can lead to the lose of hair, hide and even appendages like ears or tails.
- Groves, Colin P., and Oliver A. Ryder. "Systematics and phylogeny of the horse." The genetics of the horse (2000): 1-24.
- Hayden, Faith. "An Equine Mystery: What Killed Phar Lap?." (2010): 72-72.
- MacGregor, Arthur. "A History of Taxidermy. Art, Science and Bad Taste." Journal of the History of Collections 24, no. 1 (2012): 140-141.