And sometimes it’s the less-than-perfect version of a song that becomes a classic. Like Ella Fitzgerald’s Mack the Knife, and a few others.
I was at a Peter Gabriel concert once, about eight or nine years ago. I can’t remember which song it was, but on starting, he flubbed something, tried to recover, flubbed again, stopped, and said “ladies and gentlemen, that’s what’s known in the music industry as fucking up.” We all had a good laugh, and he nailed the song.
Except, I told that story to a boyfriend a couple of years later. He’d also seen Peter Gabriel in concert, before I had, and PG had done The Same Exact Thing, down to the joke. So, we agreed that it was a bit of friendly theatricality.
Don’t feel bad…that’s ALL my social life is currently.
Someone else mentioned Clapton on “Unplugged.”
When Paul McCartney was recording his “Unplugged,” he started singing “We Can Work It Out.” However, he started singing the wrong verse. He then stopped, said, “This is so informal, we’ll start again.” And then played the song correctly.
There’s also the recording of “You’re Only Human,” Billy Joel’s anti-suicide song. On one of the takes, he came in late on a line and kind of rushed it all in, then laughed. He was going to redo it, but (IIRC) Christie told him to leave it in, since that was the point of the song – we all make mistakes. So there it is on the album.
I went to a see Richard Zimmerman (a ragtime pianist) at a jazz club. He was really good, and by the third set everything was very casual and informal, and he said there was one piece he’d been practicing but didn’t know if he had it down well enough to play in a concert yet, but he was going to give it a try. About halfway through, in a pretty tricky sounding passage, he had to stop. He apologized, said he’d give it one more shot, and nailed it. Certain kinds of music, certain shows, sometimes you relax, be human, and let things happen.
For a famous recorded mistake, there’s I Saw Her Again Last Night by The Mamas and the Papas. After the bridge, Denny came in early with “I saw her…”, and then came in correctly with everyone else. And the group liked the way it sounded, so they left it in. (And considering the inspiration for that song, making the best of it seems somehow appropriate.) When you’re really good, even your mistakes work.
Eric Clapton: Though his blistering take of Crossroads on Wheels of Fire is regarded as golden by many of us, Clapton himself can’t bear to listen to it. He says it’s sloppy.
Artur Rubinstein: A recording of his live performance of (I can’t remember the piece) has him repeating a big section of the piece. A critic on NPR’s Performance Today griped about the fact that the CD version cropped out the mistake. Rubenstein, for those who don’t remember, was a highly regarded classical pianist.
Buddy Guy: When he played the Vogue in Indianapolis a few years ago, he got derailed in the middle of the show. He started, and gave up on, 5 or 6 songs before settling on Mustang Sally. He seemed frustrated with himself.
The Siegel-Schwall Band: The band was breaking up, and I was at their final show. This was the second show of the night, and they were plowed. They were uncommonly sloppy. A friend had seen the first show, where they were very sharp. Apparently, they got drunk between shows.
Jazz great Winton Marsalis: I can’t find the exact quote, but he said that musicians make mistakes, and that’s the part that’s jazz.
This is ancient history, but Cab Calloway’s Hi-de-ho chorus on Minnie the Moocher was improvised one night when Cab forgot the words to a verse. The audience loved it, and they kept it in. Minnie became his signature song.
The King of Ashbury Park once forgot the words to ‘Hungry heart’. But not to worry, Bruce belivers; his fans sang the entire first verse.
other famous recordings with flubs that were kept in:
someone from the Temptations says ‘pa’ a split second before the group sang the chorus of ‘Papa Was a Rolling Stone’ a second time.
The Bee Gees totally flubbed the third verse to their version of ‘Barbara Ann’.
A buddy of mine who’s an excellent guitar player in a slowly-gaining-popularity rock band told me once, “If you fuck up…do it twice and call it jazz.” Sort of what Club33 said.
David Bowie will often stop the band in the middle of a song and start over. I’m not sure if it’s due to mistakes or if he just doesn’t like the way it’s going.
I’ve sung as a member of the choir in performances with all sorts of well-known musicians: singers, conductors, instrumentalists. They are not infallible. They all can, and do, make mistakes.
Oh yes. Especially the ones who improvise a lot, a category Clapton certainly used to fit in and perhaps still does.
Gregg’s sober these days, but he’s still known to forget some lyrics now and again.
And you guys think breaking strings is unusual? Happens every other show I go to. 
The really good musicians I’ve seen have always made a wrong note sound like an intro to a new solo variation. Lindsey Buckingham does this all the time, it seems.
As for broken strings…I’ve seen real show-offs launch into blistering solos right after they’ve broken a string, just to prove that they could do it. 
I was watching a live performance of Youssou N’Dour and Annie Lennox on MTV last year (it was one of those charity-fundraising things for AIDS research). They sang Youssou’s signature song “Seven Seconds Away”. Annie was doing fine until she forgot one of the lines and started rushing into the chorus. You could clearly see Youssou getting nervous, but it worked out. It was an awkward moment, but a short-lived one.
The fact that even Annie Lennox, a goddess if there ever was one, can flub things up makes me feel better about my own screw-ups.
I saw some sad footage from one of Elvis’ last concerts in which he often forgot the words and either ad-libbed or mumbled.
I’ve been given the same advice, and since we’re both from “Music City,” that makes it official.
It’s strange what memories a question like the OP can bring back. Not quite the same, but I remember that the buglar that played Taps at JFK’s burial messed up. How weird to have that floating around in my brain after forty-two years.
The vocalist from The Kingsmen tries to come in a full bar too early on the third verse of Louie, Louie.
You can hear him sing “Me see-” before he realizes he’s screwed up.
When I saw *Jethro Tull * and Saga, I did not hear any mistakes.
But then I was pretty high…
My bass player saw Zep live (this was, as you might imagine, some years ago–like 1970 or so). His remark “I know it’s all just choices, but Jimmy Page made some pretty darned poor choices during his solos.”