Which animal sleeps standing up?

Horses can lock their legs. A mechanism of tendons and ligaments called the stay apparatus lets the animal hold its joints in position without muscular effort, so it can doze upright indefinitely and be moving within a second if something alarms it - which is the point, for a prey animal on open ground.

It cannot get all its sleep that way. Deep REM sleep requires muscle relaxation, so a horse must lie down for it, usually for less than an hour a day and often in short spells while another horse stands watch. A horse kept somewhere it does not feel safe enough to lie down will eventually become REM-deprived and start collapsing.

Others solve the problem differently. Flamingos sleep on one leg with a similar locking arrangement. Dolphins and many birds sleep one half of the brain at a time, with the corresponding eye open - a frigatebird can stay airborne for weeks, sleeping in ten-second bursts on the wing.