There have been a number of posts recently discussing various aspects of the sleeping biology of anoles (e.g., here, here, and here). Anoles spend 1/3 to 1/2 of their lives asleep, so it is not surprising that there is a small cottage industry of research papers describing where they sleep, in what position, and with whom. The most recent addition to this genre is a very nice paper on A. uniformis in Mexico, which reveals that this species is typical in sleeping on leaves with its body in line with the long axis of the leaf. The paper includes a brief, but thorough review of the literature on anole sleeping and thus is a good entrée to the literature.
A somewhat less brief review of the literature might go something like this (see Lizards in an Evolutionary Tree, from which this is drawn, for references): Many anoles sleep on leaves or on the ends of branches (although a population of A. nebulosus is known to sleep in leaf litter). The presumed function of this behavior is that any potential predator approaching the lizard will cause the branch or leaf to vibrate, alerting the lizard in time to escape by jumping into the void. This behavior may work well against such predators (as far as I am aware, no one has ever studied the efficacy of this behavior), but at least some arboreal snakes have thwarted this defense by adopting an airborne approach, stretching across from another branch to pluck the unsuspecting lizard while it still slumbers. The presence of anole remains in owl pellets suggests the existence of another threat to sleeping anoles, although another possibility is that crepuscular owls nabbed still-active anoles just as they were preparing for bed.
For many years, just about every field biologist I knew who worked on anole ecology or behavior contemplated the idea of studying whether sympatric anoles partition their sleeping sites as they do their diurnal haunts. Many workers, myself included, set out to collect the relevant data, only to discover that this was a full time project in itself. Finally, such a study was conducted in 2007 by Singhal et al. They found that for three Jamaican species (A. graham, A. lineatopus, and A. valencienni), sleeping perches are generally higher, narrower and more horizontal than diurnal perches. Females increased height at night substantially more than males, and day and nighttime habitat use was significantly different for both sexes within all three species, with the exception of A. grahami males. Despite these shifts in habitat use, interspecific differences in habitat use occurred at night, just as they did during the day.
The implication of these findings is that community and functional biologists should consider the potential importance of sleeping sites. Could species be partitioning sleeping sites as a resource? Perhaps more importantly, could the morphological differences among species represent adaptations for using different microhabitats at night, as well as during the day? The narrowness of nighttime perches is particularly notable and might make strong biomechanical demands on lizards snoozing on such perches. Another question concerns whether perch sites are chosen for their thermal properties, either at night or early in the morning, when lizards may need to raise their body temperature quickly.
Focusing on A. lineatopus, Singhal et al. found that individuals use sleeping sites that are within their diurnal home ranges. More generally, many studies have found that some individuals use the same perch repeatedly, but others do not.