From page 447 of my 1940s version of Wikipedia.

Can horses sleep while standing?

Horses have the power to sleep while standing. In 1943 Professor C.G. Winchester, member of the animal husbandry faculty at the University of Missouri, said that experiments indicate that horses rest better on their feet than on their sides. Their legs are provided with muscular mechanisms that cause them to "lock," as it were, and permit the animals to rest somewhat as if they were standing on stilts. When a standing horse is unconscious there is no direct brain control over the muscles essential to the maintenance of an erect posture.The muscles in the legs, back and chest are controlled by the reflex actions of the spinal cord. In a similar manner a bird sleeping on a swaying limb maintains a reflex balance while its consciousness is in abeyance. Horses sleeping while standing occasionally, though rarely, fall down.

More often certain muscles in the forelegs relax suddenly and the horse knuckles over onto the fetlocks and then immediately catches itself. Horses sometimes go for months without lying down. It is astonishing how little lying-down rest they require. Yet a horse left to itself will sleep, standing up or lying down, eight or ten hours a day. This is true of other Herbivora, including elephants. An Indian elephant often will feed eighteen or twenty hours and then rest and sleep only one or two. It is said that they have been known to remain standing even after they were dead. When horses lie down to sleep their eyes usually remain open or partly open and they sleep so lightly that they are awakened by the faintest sound. They seldom lie long in the same position because their great weight cramps their muscles and prevents the under lung from functioning.