S. mirabilis is known from a unique holotype specimen, which was first described by Aleksandr Grigorevich Sharov in 1971. Sharov named the species Podopteryx mirabilis, "foot wing", for the wing membranes on the hind limbs. However, that name had previously been used for a genus of damselfly, Podopteryx, so in 1981 Richard Cowen created the new genus name Sharovipteryx for the species.
The skeleton is preserved in dorsal view and largely complete, with the bones still articulated and impressions of some of the integument. But part of the pectoral girdle is missing and part is still encased in stone.
In 1987, Gans et al. published a revised description: they found that the patagium did not extend to the forelimbs. Their experiments with models showed that the reptile could glide with its uropatagium and stabilize its glide by changing the angles of its forelimbs to provide an aeronautic canard or by bending its tail up or down to produce drag.
In 2006, Dyke et al. published a study on possible gliding techniques for Sharovipteryx. The authors found that the wing membrane, which stretched between its very long hind legs and tail, would have allowed it to glide as a delta wing aircraft does. If the tiny front limbs also supported a membrane, they could have acted as a very efficient means of controlling pitch stability, very much like an aeronautic canard. Without a forewing, the authors find, controlled gliding would have been very difficult. Together with the canards on the forelimbs, these anterior membranes may have formed excellent control surfaces for gliding. Unfortunately, the area around the forelimbs was completely prepared away in the only known fossil, destroying any possible trace of a membrane there.
Sharovipteryx is generally agreed to belong to a group of early archosaur relatives known as the protorosaurs (or prolacertiformes). One researcher, David Peters, has suggested that protorosaurs like Sharovipteryx were closely related — or perhaps even ancestral — to pterosaurs, although this classification has not been supported by other scientists. Unlike pterosaurs, the main flight membrane of Sharovipteryx was stretched between long back legs rather than its very short front limbs.
If Sharovipteryx was a relative of pterosaurs as Peters suggests, then its membrane may have stretched to its front legs, or it may have had a separate membrane joined to its front limbs alone. Peters claimed to have found a secondary membrane between the thighs and the trunk by tracing photographs of the fossils specimen, and claimed to have traced the fingers, which other scientists who study the fossils reported were not preserved. According to Peters, the fingers show similarities to Cosesaurus and Longisquama and to a lesser extent, pterosaurs. However, Peters' photo tracing method for studying fossils has been criticized as "fantasy" by prominent pterosaur researcher Chris Bennett, who vehemently denounced the methodology. Palaeozoologist Darren Naish has explained why palaeontologists believe that Peters' claims are unreliable: Peters works from photographs, not from the fossils themselves; he traces as parts of the anatomy in place artifacts which may be products of deocomposition, random variations in sediment, or even pixillation in zoomed-in images; some of the features may be the result of the mind's tendency to make patterns; some of his discoveries imply offspring that are infeasibly small; his radically different taxonomies are constructed by adding features that cannot be conclusively seen and measured; those taxonomies differ from existing taxonomies that are well supported by morphological evidence.