Horses doze standing up, but do not go into an actual sleeping state. Their need for real sleep is much less than ours, a couple or three hours out of 24, and not all at once, is sufficient. This deep sleep is accomplished laying down.
In a herd, only one or two horses will lay down at a time. A few others will doze, and the rest will be grazing or keeping watch - in the picture below, the bright bay horse is keeping watch while his dark bay/brown friend soaks up some rays:
Contrary to popular mythology, cows do NOT sleep standing up. They can not “lock” their legs. One cannot “tip” cows for this reason.
Even if cows (an alert prey animal) slept in a standing position (which they do not), the idea that a drunken human being could sneak up on them and topple over a 1000 lb animal “braced” for sleep is absurd.
Cows lay down much more frequently than horses. They typically lay down when chewing their cud and this occurs a few times a day. Unlike horses, cows have a form of day-care. Several cows will leave all their babies with one cow to oversee.
Now foals, especially really young ones, can spend a good part of their day laying down. They’re awfully cute to watch lay down when they’re less than a week old - they don’t have their legs really figured out yet, and it can take 5 minutes or more for them to lay down. They can get up REALLY fast, but can take a long time going down.
Once on the ground, they’ll often start with their legs tucked around themselves - this is the position from which they can rise quickest. Then as fatigue overcomes them, they’ll lower their nose to the ground, their legs will untuck, and they’ll slump over on their sides - especially if the sun is nice & warm.
I had my horse at a horse show one week, and nearly died laughing when I came upon my horse sound asleep in her stall in the same position as this miniature horse (without the couch, of course. My horse was square on her back in the middle of the stall): The night watchman told me she slept that way every night!