Wings are those limbs of a flying creature that provide the lifting force.  Those of birds are feathered, those of bats membranous; the feathered wing is the default type, to be used unless otherwise specified.  All wings are displayed by default.
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A wing may be used as a single charge; this usage dates from c.1295, in the English arms of Peek [ANA2 556].  Both dexter wings and sinister wings are found in period armory.  The mundane default has varied between countries and times; the Society default is the dexter wing. 
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In German heraldry, the single wing may “terminate in a hand” (often maintaining a sword, as in the illustration), with the hand opposite the wing’s severed end; it’s found in the arms of the Dukes of Calabria, 1413 [Conz.Const. xcviii].  This is distinguished in blazon from a hand or claw “issuant from a wing”, where the hand issues from the wing’s severed end; it’s found in the arms of the Marquis de Vilena, c.1370 [Gelre 62v].  (There is also an example of a wing terminating in an eagle’s head, in the arms of von Ernberg, 1605 [Siebmacher 103].)  These variations are always blazoned.
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Wings are also found in pairs, with a dexter and a sinister wing, frequently conjoined.  (The difference is subtly blazoned:  two separate, dexter wings would be blazoned “two wings”, while a dexter wing and a sinister wing would be “a pair of wings”.)  This usage dates from c.1285, in the arms of Derneford [ANA2 555].

A pair of wings may be conjoined: this was considered artist’s license, as the same arms could be drawn either with the wings conjoined or separate. A pair of wings displayed and conjoined may also be blazoned a “vol”, the French term for the motif; this is seen in the arms of von Hohenfels, 1606 [Siebmacher 140]. If the conjoined wings are displayed with tips inverted, they are known as “wings conjoined in lure”, as in the arms of Jane Seymour, d.1537 [Woodcock & Robinson pl.19].

Finally, of Society-unique charges, we find the “set of seraph’s wings”:  six wings conjoined, arranged as if attached to a seraph.

Bat’s wings are much less common in medieval armory than bird’s wings:  Your Author knows but a single example, the badge of Daubeney, Earl of Bridgewater, d.1548 [HB 81].  For related charges, see lure.

The Baron of the Angels bears as a badge:  A set of seraph’s wings Or.

The College of Cathanar bears:  Vert, a sinister hawk’s wing argent and in canton a laurel wreath Or.

Matill of Windkeep bears:  Purpure, three sinister wings argent.

Brioc Morcannuc bears:  Azure, a vol Or.

Herman Mandel bears:  Barry and per pale sable and Or, a wing terminating in a hand maintaining a sword all within a bordure gules.

Etienne Michel de Calais bears:  Argent, in pale three pairs of bat-wings conjoined gules.

Gustavus von Goslar bears:  Or, an eagle’s dexter wing terminating in an eagle’s head sinister facing sable, a chief rayonny gules.