Tom Waits is sometimes accused of this. He seems well aware of it, as at least two of his songs - "Let Me Get Up On It" from Bone Machine and "Dog Door" from "Orphans: Bawlers, Brawlers & Bastards" - have no printed lyrics, and the former may not include any actual words at all.
MadTV did a spoof sketch on this, featuring the boy from The Calling, then featuring the guy from Creed accusing him of stealing the way he sings. Then "the guy from Pearl Jam" Eddie Vedder shows up to accuse them all of stealing from him. It then ends with Ray Charles (who was still alive at the time) showing up and claiming they all stole from him first.
Though Bob Dylan's songs themselves aren't particularly unintelligible, he will always be portrayed as such in parodies. A particularly notable example is the last verse of Eric Bogle's "Do You Know Any Dylan?" which simply doesn't have any consonants at all. Probably more accurate to say that all the words themselves are perfectly legible, but the whole makes less sense than the sum of its parts.
Ozzy Osbourne. Oddly enough, understanding his songs isn't too hard, but some people have trouble understanding him when he talks.
He once made a video where he drove this one into the ground. In reality (I.E. not that video) he is mostly intelligible, though he does mumble sometimes. Neurologically, people use different areas of the brain's language center for speaking and singing. That's why people who stutter (such as country singer Mel Tillis) can sing without doing so, and why Ozzy's much easier to understand when he sings.
In Conversations with Tom Petty, Petty himself gives an example: "When Linda Ronstadt did 'The Waiting,' she phoned me, because she couldn't figure out all the words. It was the line 'No one could have ever told me about this...' She couldn't figure out what the hell it was. And I thought, 'Damn, I've got to try to enunciate better.'"
Petty also mentioned in a 1979 article that producer Jimmy Iovine told him that he sang like he had a mouth full of food.
Kurt Cobain's half-stoned mumbling in "Smells Like Teen Spirit" is the entire reason for the "Weird Al" Yankovic parody, "Smells like Nirvana." According to Al, when he asked for permission to make "Smells Like Nirvana," Cobain asked, "It's not going to be about food, is it?". When Cobain learned that it would be about how no one could understand his lyrics, he was quite amused.
When John Flansburgh of They Might Be Giants did his own accordion cover, he just sang "Mnyeh mnyeh whooah" noises. It sounded remarkably convincing.
And Japanese beatboxer Dokaka's cover appears to lampshade this; the only actual words (in any language) is the phrase "buona sera." Until you listen to the rest of his material and realize all his songs involve scatting over vocal melodies, regardless of the source's vocal legibility.
The opening theme to the Hellsing TV series, "World Without Logos," has lyrics that appear to be sung by someone in a later stage of aphasia.
Early R.E.M. albums (Murmer was named for this phenomenon) had Michael Stipe singing in a slurred, "mumbly" style, which was one of the reasons they stood out as much as they did in the divided 80's. They broke away from this approach with their 1986 album Lifes Rich Pageant, coinciding with both a change in producer and shift in lyrical focus from cryptic poetry to sociopolitical musings. Stipe has gone on record saying that "the early albums didn't have lyrics", a reference to his penchant for taking a stream-of-consciousness approach to lyric writing and singing at the time.
My Bloody Valentine are famous for burying their vocal legibility under walls of guitars and asynchronous harmonies. Unless you have the original mix tapes on hand, good luck figuring out the lyrics. Shoegaze in general is infamous for this, due to the tendency to bury vocals in the wall of guitars.
Ben Sotto of Heavenly, a French band that releases and tours mostly in East Asia, has a masterful way of ignoring all English emphasis, syllablery and pronunciation, never seeming to quite match up to any other human being's interpretation of the language. Watching their first video single while trying to read along with it is an interesting exercise.
Brian Johnson of AC/DC. Just listen to the album version of "Thunderstruck." His destruction of his vocal chords in the mid-'80s (which isn't surprising, given his, erm, "singing" style) compounded this quite a bit.
Subverted by a track called 11h30 by DANGER. It has yet to have any correct lyrics sheet for it, but sounds like human speech. When asked if the song was meant to be unintelligible, the artist said that it was made up of fractured and mixed-up syllables spoken in French and was not meant to be 'unintelligible', but more 'whatever the listener believes it is supposed to be'.
Barry Cryer and Ronnie Golden's parody blues singer Mumblin' Jack Spanner. He achieved his unique style after he lost his teeth. In a poker game.
Paul Westerberg, during many of The Replacements' early live shows. This wasn't really a result of how he sings; it was mostly because of how drunk he was.
Black Metal is often even worse at this, due to the (often intentionally) under-produced and lo-fi recording style, and due to many bands choosing to sing in the language of the country they are from, as opposed to English, or often, when English is used, extremely thick accents, not only make what is being said nearly impossible to understand, but it is often difficult for listeners to determine what language the songs are being sung in.
Grindcore in general. Especially frustrating since it's genre built on quirky lyrics. When deprived of both a lyric sheet and a track listing, Anal Cunt become more of a prank than a joke band.
Some of the more obscure Metalcore bands (or bands that started out metalcore in their early days) are notorious for invoking just about every metal trick in the book to garble their vocals. Try deciphering more than a single line per song on As I Lay Dying's first album Beneath the Encasing of Ashes without a lyric sheet on hand.
German singer Herbert Grönemeyer. Less so in the studio records, very much so in live performances. Often parodied.
Some of Slipknot's earlier work suffers from this at times. Namely Corey's fast-paced rapping that was exclusive to their self-titled album. Then there's their self-released debut Mate, Feed, Kill, Repeat in which Anders Colsenfi handled lead vocals, often in the tried-and-true "cookie monster" style so often associated with this trope.
Not only is Korn frontman Johnathan Davis a major offender with his psychopathic vocal style, but the band further compounds the matter by offering no lyric sheets. Go look online for lyrics, you'll find no two websites offer anywhere near the same results.
Sly Stone is clearly stoned out of his gourd throughout "There's A Riot Goin' On."
"Louie Louie" by the Kingsmen is so unintelligible that after a complaint that it was obscene, the FBI, after two years of investigation, concluded that the recording could not be interpreted, that it was "unintelligible at any speed," and therefore the Bureau could not find that the recording was obscene. Your tax dollars at work with the FBI laboratory... And yet, somehow they managed to completely miss what is often the most overlooked f-bomb in music history. About a minute into the song, you can hear drummer Lynn Easton accidentally hit his sticks together, and he swears in frustration and what he says is quite clear - if you know what to listen for. See for yourself: @ :55 seconds in.
Dave Mustaine's vocals on the first Megadeth album Killing is My Business... and Business is Good is a garbled mess thanks to a mumbled delivery made even worse by a blurry production job that drowns it in an endless sea of reverb. Peace Sells... but Who's Buying? isn't much better in this regard. So Far, So Good... So What! shows substantial improvement, however, despite Dave being at the apex of his drug addiction at this point, as well as some of the reverb that plagued Killing is My Business making a comeback. By Rust in Peace (his first record after getting clean), Mustaine's singing is pretty clear, with only a few "break out the lyric sheet" moments. Fortunately, the remasters of the early albums go a long way toward fixing the problems with those early vocal tracks.
The Butthole Surfers fall into this trope every now and then in their early career deliberately, thanks to lead singer Gibby Haynes' "Gibbytronix system. He claimed at one point it originated to mask the fact that he couldn't sing, but now that that's changed, he just uses it to enhance his vocal effects.
"The End" by The Doors from The Doors: Jim's screaming during the line "mother... I want to... rape you!" in "The End" is so garbled that it is impossible to make out what he is shouting, therefore it sounds more like "mother... I want to... murder you!".
Done intentionally in the song "Little Demon" by Screamin' Jay Hawkins which has a chorus section that is unintelligible gibberish. The joke is that the gibberish is the Devil speaking, and "you gotta be real cool to hear the words he said.", implying that the listener (you) is NOT cool.
Ariana Grande is well-known for this, despite people noting that her actual vocals are fantastic - it's just that her enunciation is awful. She's also quite good at doing impressions of other singers, and sings much more clearly when mimicking them, leading people to joke that she's only coherent when she's pretending to be someone else.