Little Hawk said: "This seems to be mainly a U.K. problem, judging by the posts."

Well, here on the west coast of the US, I participate in several different "jam sessions" and "song circles" ("singaround" doesn't seem to be a term used in these parts). In jam sessions, it's expected that people will play and/or sing along -- even taking turns with instrumental breaks, if they like.   In those, it's not uncommon for the fiddle players, e.g., to request a more fiddle-friendly key than, say, A-flat.   But they generally do that before you start playing.
"Song circles" are generally called that because you sit in a circle (as opposed to having someone on stage. In one mode, people take turns either performing a song, leading one, or requesting one. Leading songs is common, and people tend to assume it's OK to do instrumental backup, in addition to singing along, unless asked not to.   But it's usually also understood that you don't drown out the singer or throw them off with your rhythm, etc.   So it's normally not thought of as rude to jump in with your guitar (or vocal harmony), but you're expected to be sensitive to nonverbal cues, etc.   Occasionally, though, I do find that too many instruments mess up a song I'm doing or that the whole group seems bent on singing my song faster or slower than I want -- or not allowing me to pause between musical lines as I see fit.   But, yeah, it's not Carnegie Hall, and some of the snags do come from some people knowing less about music (e.g., not understanding the concept of a "rest" in music notation).   



But Bee mentioned having people start a whole new country song while she was still singing her folk song in a "country" singaround.
In "song circles," we sometimes go into "chaos mode," meaning no more formally taking turns in a circle. And some people do tend to start more than their share of songs, but it's still understood that you don't jump into your new song before someone else finishes theirs.

I've had that happen to me, but due no doubt mostly to the ambient noise and free flow of beer in a chantey/folk song sing.   It was at an afterparty at NW Folklife Festival a few years back.   Several times, I managed to get a few bars of a song out in the wink of an eye between the end of the last song and someone else's diving in.   My voice is soft, but not inaudible. Every single time I managed to start a song, I'll be damned if one of the big-voiced chantey singers didn't start bellowing out a different song right on top of mine.

I just gave up on that informal gathering (after singing along with quite a few), because of the atmosphere.   It was a lost cause.   But I've actually had similar things happen once or twice in much smaller, quieter settings.