- Sporadically in Classical Greek and Latin, as a rhetorical technique for both prose and poetry, rather than a basis of verse: Homeoteleuton.
- Systematically in Arabic and Chinese, but I don’t know much about them.
- In Europe, rhyme emerges as a structural feature of verse (as opposed to an occasional device) in the Late Middle Ages. I see from Wikipedia (Rhyme) that the likely source was Arabic poetry. Once it got started in Europe, it migrated from place to place; in Greek, it turns up in the 14th century (Stephanos Sahlikis), under Italian influence.
Different languages differ in their tolerance of “poor” rhyme, or even their definition of it (French regards it as the final vowel; Portuguese and Esperanto as the same inflection); their tolerance of imperfect rhyme; their pursuit of identical rhyme, and so forth. The phonology of the language determines how much rhyming is possible, and therefore what kind of rhyming is desirable.
Literary fashion determines whether rhyme is still considered current or old-fashion in high literature (as opposed to song lyrics, which are arguably the mainstream modern form of poetry at least in the Anglosphere). Russian still likes rhymes; in English literary circles, rhyme is now considered a regression. (The libretto of Nixon in China was labelled rhymed, when in fact it’s largely assonance.)