From c. 1400 as "without curvature or projection." Sense of "prosaic, dull" is from 1570s, on the notion of "featureless, lacking contrast." Used of drink from c. 1600; of women's bosoms by 1864. Of musical notes from 1590s, because the tone is "lower" than a given or intended pitch. As the B of the modern diatonic scale was the first tone to be so modified, the "flat" sign as well as the "natural" sign in music notation are modified forms of the letter b (rounded or square).
Flat tire or flat tyre is from 1908. Flat-screen (adj.) in reference to television is from 1969 as a potential technology. Flat-earth (adj.) in reference to refusal to accept evidence of a global earth, is from 1876.
1801, "a story of a house," from Scottish flat "floor or story of a house," from Old English flett "a dwelling, hall; floor, ground," from Proto-Germanic *flatja-, from suffixed form of PIE root *plat- "to spread." Meaning "floor or part of a floor set up as an apartment" is from 1824. Directly from flat (adj.) come the senses "level ground near water" (late 13c.); "a flat surface, the flat part of anything" (1374), and "low shoe" (1834).