Binocular fusion relies on matching points in the two eyes that correspond to the same physical feature in the world; however, not all world features are binocularly visible. Near depth edges, some regions of a scene are often visible to only one eye (so-called half occlusions). Accurate detection of these monocularly visible regions is likely to be important for stable visual perception. If monocular regions are not detected as such, the visual system may attempt to binocularly fuse non-corresponding points, which can result in unstable percepts. We investigated the hypothesis that the visual system capitalizes on statistical regularities associated with depth edges in natural scenes to aid binocular fusion and facilitate perceptual stability. By sampling from a large set of stereoscopic natural images with co-registered distance information, we found evidence that monocularly visible regions near depth edges primarily result from background occlusions. Accordingly, monocular regions tended to be more visually similar to the adjacent binocularly visible background region than to the adjacent binocularly visible foreground. Consistent with our hypothesis, perceptual experiments showed that perception tended to be more stable when the image properties of the depth edge were statistically more likely given the probability of occurrence in natural scenes (i.e., when monocular regions were more visually similar to the binocular background). The generality of these results was supported by a parametric study with simulated environments. Exploiting regularities in natural environments may allow the visual system to facilitate fusion and perceptual stability when both binocular and monocular regions are visible.