Meaning "examine, inspect" (usually prior to robbing) is from 1915, American English slang, perhaps from the notion of giving a place a look on all sides. Compare technical case (v.) "cover the outside of a building with a different material" (1707), from case (n.) "external portion of a building" (1670s).
early 13c., "what befalls one; state of affairs," from Old French cas "an event, happening, situation, quarrel, trial," from Latin casus "a chance, occasion, opportunity; accident, mishap," literally "a falling," from cas-, past-participle stem of cadere "to fall, sink, settle down, decline, perish" (used widely: of the setting of heavenly bodies, the fall of Troy, suicides), from PIE root *kad- "to fall."
The notion is of "that which falls" as "that which happens" (compare befall). From its general nature, the word has taken on widespread extended and transferred meanings. Meaning "instance, example" is from c. 1300. Meaning "actual state of affairs" is from c. 1400. In law, "an instance of litigation" (late 14c.); in medicine, "an instance of a disease" (late 14c.).
The grammatical sense, "one of the forms which make up the inflections of a noun" (late 14c.) also was in Latin, translating Greek ptōsis "declension," literally "a falling." "A noun in the nominative singular ..., or a verb in the present indicative ...,
is conceived as standing straight. Then it falls, or is bent, or
declines into various positions" [Gilbert Murray, "Greek Studies"]
U.S. slang meaning "person" (especially one peculiar or remarkable in any way) is from 1848. Meaning "incident or series of events requiring police investigation" is from 1838. In case "in the event" is recorded from mid-14c. Case-history is from 1879, originally medical; case-study "study of a particular case" is from 1879, originally legal; case-law "law as settled by previous court cases" is from 1861.
"receptacle, box, that which encloses or contains," early 14c., from Anglo-French and Old North French casse (Old French chasse "case, reliquary;" Modern French châsse), from Latin capsa "box, repository" (especially for books), from capere "to take, hold" (from PIE root *kap- "to grasp").
Meaning "outer protective covering" is from late 14c. Also used from 1660s with a sense of "frame" (as in staircase, casement). Artillery sense is from 1660s, from case-shot "small projectiles put in cases" (1620s). Its application in the printing trade (first recorded 1580s) to the two shallow wooden trays where compositors keep their types in compartments for easy access led to upper-case for capital letters (1862), so called from its higher position on the compositor's sloped work-table, and lower-case for small letters.
The cases, or receptacles, for the type, which are always in pairs, and termed the 'upper' and the 'lower,' are formed of two oblong wooden frames, divided into compartments or boxes of different dimensions, the upper case containing ninety-eight and the lower fifty-four. In the upper case are placed the capital, small capital, and accented letters, also figures, signs for reference to notes &c.; in the lower case the ordinary running letter, points for punctuation, spaces for separating the words, and quadrats for filling up the short lines. [The Literary Gazette, Jan. 29, 1859]