Horses don't sleep much, maybe 3 or 4 hours out of every 24. When they do nap, it's usually while standing. On a nice day, they'll lie down in the sun the same way a deer will, with their heads up. If they doze off, their noses dip toward the ground. Like deer, horses are prey animals, and have to be ready to jump up and run if predators appear. Consequently, I rarely catch my horses lying flat-out on the ground, completely vulnerable and oblivious.
Except for Code Red. When he naps, he's serious about it. Dead serious. I've learned not to get nervous if the dead body in the pasture is Code Red, because he's just enjoying a nap.