His transformation phrase is "I summon within, the Power of Wind!" The rhyme just doesn't sound right to begin with, and it's made doubly painful by sticking out like a sore thumb from the others' transformation phrases, which simply have them declaring "Power of [x]!"
"Wetter Works" has three of these, each worse than the last. He starts by resorting to "fella" to rhyme with "umbrella". A short time later, he follows up with the rather oblique "tongue/song". Then, after the Monster of the Week is beaten, Artie calls Brain a "genius", which Rhymey attempts to rhyme with "all of us". Ow.
"Double Trouble" has the doublet "cocky/quickly". Incidentally, this one also has Artie supplying the first line and Rhymey attempting to rhyme with his own.
In "Three to One", he gets the gem "My sword's dent/Twisted and bent". This one's made painful not by the rhyme itself, but for resorting to glaringly incorrect grammar just to try and make the rhyme work, namely using "dent" as a past-tense verb as opposed to "dented".
In "A Visit to Equestria", Rhymey tries to rhyme "truly" with his own name. Later he resorts to the Perfectly Cromulent Word "bite-full" to rhyme with "delightful", followed by "tradition/generation".
During the infiltration of the Nightmare Moon-dominated Equestria in "Reunions and Unions", he gets "wondrous/see us". Clunky as it is, the original My Little Unicorn uses the even more cringeworthy "for us/see us". Before that, he gets to rhyme "ruin" with "doin'".
Act 6 of Starfleet Humans has the human Rhymey butchering grammar for the sake of rhyme: "I cant ever confess to Fluttershy/A broken mess will always be I."
Contrary to what Rhymey's dialogue from Act 2 of Starfleet Events tries to suggest, "awry" is pronounced with an "I" sound and does not rhyme with "sorry".
It's not just limited to Rhymey, either; "Careful What You Wish For" in season 3 has a genie who also speaks entirely in rhyming couplets, one of which attempts to rhyme "ago" with "go", which is just as clunky as it appears in context.
The D-Man Knights' Villain Song has them rhyming "D-Man" with "He-Man", "She-Man, "Be-Man", and "C-Man". It's evident that the writer really, really couldn't come up with anything better to rhyme with.
Taken Up to Eleven in My Brave Pony: Starfleet Magic II, where there is an entire chapter written in rhymes. Ironically, it may be the best chapter in the entire story, despite it copying an episode of Arthur.