Old English sciete (West Saxon), scete (Mercian) "cloth, covering, towel, shroud," from Proto-Germanic *skautjon-, from *skauta- "project" (cf. Old Norse skaut, Gothic skauts "seam, hem of a garment;" Dutch schoot; German Schoß "bosom, lap"), from PIE root *skeud- "to shoot, chase, throw" (see shoot (v.)).
Sense of "piece of paper" first recorded c.1500; that of "any broad, flat surface" (of metal, open water, etc.) is from 1590s. Of falling rain from 1690s. Meaning "a newspaper" is first recorded 1749. Sheet lightning is attested from 1794; sheet music is from 1857. Between the sheets "in bed" (usually with sexual overtones) is attested from 1590s; to be white as a sheet is from 1751. The first element in sheet-anchor (late 15c.) appears to be a different word, of unknown origin.
"rope that controls a sail," late 13c., shortened from Old English sceatline "sheet-line," from sceata "lower part of sail," originally "piece of cloth," from same root as sheet (n.1). Cf. Old Norse skaut, Dutch schoot, German Schote "rope fastened to a sail."
This probably is the notion in phrase three sheets to the wind "drunk and disorganized," first recorded 1821 (in form three sheets in the wind), an image of a sloop-rigged sailboat whose three sheets have slipped through the blocks are lost to the wind, thus "out of control." Apparently there was an early 19c. informal drunkenness scale in use among sailors and involving one, two, and three sheets, three signifying the highest degree of inebriation; there is a two sheets in the wind from 1815.
[1821+; fr the fact that a drunken person is as helpless and disorganized as a sailboat with its sheets, that is, with its sails, flying, and hence its course and movement entirely out of control]