The whole question started by this thread is very much a "horses for courses" one, but the ensuing debate has quite rightly highlighted some of the potential pitfalls and inequalities of whichever method (or combination) is adopted.

The duos/etc matter, especially, is an interesting one, and it is especially prone to malpractice.
From experience, how I would see it operating at its fairest is by determining whether those performing are defined as an "act" in its (their) own right:
Assume there's a (fictitious) duo, let's call them John and Jane. Jane's a singer-songwriter, always sings her own songs or a few covers, whatever, but always with John accompanying, John not being a singer or solo performer in his own right. So surely it's fairest to regard the duo as one named "act" on the bill, and thus they get one song per turn round the room, just like each other named "act".
Otherwise, too, if it's one of the familiar situations where there are so many participants that it takes two hours just to get round the room once, it's blatantly unfair on the patient solo performer who may not get to sing again at all...
Sometimes John and Jane are augmented by a fiddle player, Steve. But again that is essentially the same "act". Now if Steve, who happens also to be a gifted session musician, decides to play a set of tunes solo in his own right, then that would legitimately count as a separate "act" with his own "turn" at the singaround or session.
If, however, John and Jane are each solo singers with a different repertoire of an individual or contrasted nature (Tom & Barbara's particular scenario is relevant here), then the case can be made for "one song each" per turn round.
Similarly with a harmony-based singing trio (let's think of one like Craig Morgan Robson, each member of which is a superb solo singer anyway), who might in a singaround situation choose to sing together as an "act" or else choose to sing one song each as a solo - but they wouldn't expect to get to perform two or three songs each "turn" as an automatic right.
Any "act" worth their salt would take a similarly professional and relatively self-effacing attitude, I feel sure.

IMHO, and increasingly, too many "organised" events that are billed as singarounds or singing sessions or "big sings" are being turned into a succession of glorified "guest spots", whether by accident or design (or just plain inefficient MC-ing) - but that's another debate...

Generally I'd agree with previous correspondents on this thread, ie. that the problem lies with the "acts" (or groups of mates) who come in mob-handed, armed with four or five instrument cases (say), they may undertake excessive tuning/getting ready, and then (quite arrogantly) fully expect to be allocated more than one song per "round"... those who use the singaround situation as a strategic "play, then cut and run" and/or a means of getting an easy booking (!)... those who are not prepared to stay and listen to anyone else...

Parity is a wonderful thing. And so is the serendipity of a friendly, well-run and non-competitive singaround.