The singarounds I've enjoyed most have generally been completely unaccompanied ones. This gets rid of the problems of inappropriate accompaniments, out-of-tune instruments (buy not, alas, tuneless voices), long instrumental episodes that break the flow of the song, and the instrumental noodlers who hog the bandwidth and make it difficult for anyone else to get started.

I can see that some singers don't like to "push in", so some sort of turn-taking is usually best. I find it makes me jumpy not knowing when my turn is coming, so I prefer not to be "nominated" without warning. The system where you pass some object around the room, and take your turn when you've got it, (like the stick in Sidmouth Middle Bar) can work well, as long as the singer passes it on to someone else as soon as their own turn begins. That gives the next singer time to think of something that's suitable to follow on from the current song.

Marje