Whether or not a person can sleep standing up really depends on a person's definition of sleep. A number of people who must remain standing while performing repetitive work claim they have fallen asleep on their feet, but the experience is often more of a temporary mental shutdown than physical sleep. Under most conditions, it is virtually impossible for a person to sleep while standing for hours without any outside support at all.
The difficulty does not lie in the mental aspect of falling asleep, but more in the physical, at least for humans. People simply cannot lock out their supportive joints long enough to allow for hours of uninterrupted sleep while remaining vertical. The reason a four-legged animal such as a cow or horse can sleep standing up is anatomy. Cows and horses have the innate ability to lock their knee joints, allowing them to place all of their weight on their legs as they sleep. They stay standing up in order to maintain the option of a hasty exit if threatened.
Humans, on the other hand, have no such abilities. When a human tries to sleep while standing, he or she generally fails within a few minutes. Sleep involves a general relaxation of the muscles, including those responsible for keeping a person standing upright. Because the would-be sleeper's mind has largely shut down, it is not providing all of the subtle and not-so-subtle adjustments that keep standing humans in balance. In essence, once the brain, joints, and muscles stop working to keep a person upright, it's time to meet the floor.
As far as sleep positions are concerned, even if it were possible to sleep standing up, constant stress on the spine would most likely create a new level of pain for the sleeper the following morning. A supportive body brace might overcome the relaxed joint situation, but other pressure and pinch points would make sleep just as difficult. Attempting to sleep while standing up would appear to defeat the restorative purpose of sleep in general, although a short mental catnap might still be possible under the right circumstances.