Hi Pamela, good to hear you're busy with your research. This is the kind of thing I've always thought I should set time aside for, but never got round to. What I would do is to pick a song with a lot of variants, then try to correlate the range of melodic variation with their geographical spread. Often you will find that two variants from the same community are very similar (although sometimes they can be quite different), but the same song collected 20, or 100 miles away might show more variation. You might also be able to find a version of the same song collected in the same area say thirty years later and see how much it had changed.

An obvious place to start would be Cecil Sharp's collection from the Appalachians, all of which is available online. Within it you can find multiple variants of the same song, and could then compare (say) the variation between versions from NC, KY and TN, as against variations within a small area like Madison Co., NC., where Sharp really concentrated his early efforts.

On the question of tunes that carry several songs, Will Fly has already pointed out that 'Dives and Lazarus' has done service for a whole number of textually unrelated songs from the British Isles (Cecil Sharp found it in Virginia, come to that). The tune often referred to as 'Villikins and his Dinah' is another common one, and in 'The New Penguin Book of English Folk Songs' you can find three songs sharing the tune best known from 'Flash Company'.

As far as I know, no-one has yet invented software that will analyse points of similarity and difference between melodies. Now that would be a handy thing.