1590s, from up (adv.) + stairs (see stair). As an adjective from 1782. The noun is first attested 1872. Adjectival meaning "characteristic of upstairs life" (in private rooms of a household, as opposed to servants' quarters) is recorded from 1942.

He [Halifax] had said he had known many kicked down stairs, but he never knew any kicked up stairs before. [Gilbert Burnet, supplement to "History of My own Time," from his original memoirs, c. 1697]

"down the stairs, below, to or on a lower floor," 1590s, from the prepositional phrase; see down (adv.) + stair. As an adjective, "pertaining to, relating to, or situated on the lower floor of a house," by 1819. As a noun, "the downstairs part of a building," by 1843.

"optical illusion of objects reflected in a sheet of water in hot, sandy deserts," 1800, in translations of French works, from French mirage (1753), from se mirer "to be reflected," from Latin mirare (see mirror (n.)). Or the French word is from Latin mirus "wonderful" (see miracle). The similarity to Arabic mi'raj has been noted, but the usual sense of that word is "ladder, stairs; climb, ascent," and the resemblance appears to be coincidental. The standard Arabic for "a desert mirage" is sarāb. The figurative sense of "deceptiveness of appearance, a delusive seeming" is by 1812. The phenomenon is produced by excessive bending of light rays through layers of air of different densities, producing distorted, displaced, or inverted images.

[standard of measure or estimation] late 14c., "series of registering marks; marks laid down to determine distance along a line," (in Chaucer's description of the astrolabe), from Latin scala "ladder, flight of stairs," from *scansla, from stem of scandere "to climb" (see scan (v.)). 

The noun in the classical Latin sense is rare, though Middle English had it as "ladder used in sieges" (c. 1400). The meaning "succession or series of steps ascending or descending" is from c. 1600; that of "standard for estimation" (large scale, small scale, etc.) is from 1620s.

The musical sense of "definite and standard series of tones within a certain range," typically an octave (1590s), and the meaning "proportion of a representation to the actual object" (1660s) are via Italian scala, from Latin scala.

"to climb (a wall) by or as by a ladder; attack with scaling ladders," late 14c., scalen, from Latin scala "ladder, flight of stairs," from *scansla, from stem of scandere "to climb" (see scan (v.)).

Middle English scale, "ladder used in sieges," is attested c. 1400, from the Latin noun. The verb in general and figurative use (of mountains, heights of pleasure, etc.) is from 16c.

Via scale (n.3) "standard of measure or estimation" comes the meaning "measure or regulate by a scale" (1798), the sense of "draw, project, or make according to scale" (by 1885), and scale down "cut or decrease proportionally in every part" (by 1887). Related: Scaled; scaling.