As with a lot of things rhyming in poetry seems to be subjective to current fashion. I've always been under the impression the Romans didn't like rhyming poetry. The poetry I learned growing up did rhyme. But modern poetry usually doesn't seem to rhyme.
So my question is. When and why was it deemed fashionable to rhyme in poetry? Or wasn't it as much of an era and more of a personal thing?
You would have to name a specific poetic tradition, and generally or implicitly, the poetry of a specific language, in order to address the coming and going of rhyme as a preferred poetic structure or style. I'll address it in the English context, in any case, as that's what we're currently writing in, on this site.
Rhyme's relationship to English poetry has always been a tad fraught. For the first 600 years of English poetry, it was not the common practice, with alliterative verse being preferred. Ultimately, English yielded to Latin rhyme, over the course of the period from Late Middle English (when Chaucer becomes the voice of Rhyme Royal) to Early Modern English (when the Early of Surrey introduces the English Sonnet, as well as Iambic Blank Verse).
This period from 1400 to 1600 is the heyday of rhyme in English, when it is explored and innovated upon. Alliterative verse is almost completely expurgated from the tradition. And the popularity of rhyme persists into the late 19th century. But by that time, some critics are beginning to have their doubts. As much as Swinburne and Tennyson are appreciated for the wonders they work with the intractably diverse syllabary of English, some begin to contend that the rhyme which so well served Italian and Latin simply does not well serve English, and the problem was never solved.
From the beginning, concessions had to be made. The Petrarchan Sonnet, with its ABBA ABBA rhyme structure, simply demanded more rhymes than English could reasonably be expected to provide, in all its phonotactic diversity. Even where such rhymes could be found, this excess of rhymes sounds rather unnatural to English, where a rhyme is so seldom incidental. The Shakespearean Sonnet was a happy middle ground. The alternative to this refashioning of rhymed poetic forms was the blank verse which Surrey, Shakespeare and Milton are otherwise famous for.
In any case, by the time of Swinburne and Tennyson, as I say, some critics express what almost sounds like pity, for these men tasked with applying their genius to the objective of saving English rhyme from itself.
But rhyme by no means died out, in English poetry. More so, it yielded to the diversity of poetic forms that the language accommodates.
Are we talking rhyming throughout the whole thing, or even the inclusion of phoning couplets? Are we making distinctions between poetry & the introduction of prose? Many poets fit their works into structural forms, eg, sonnets etc. this informs the inclusion of rhymes. Many poets have also experimented with prose, which quite often don't rhyme but are quite melodic & often confused with poetry.