I've got a book: "Folk Songs of Old Hampshire" edited by John Paddy Browne which has got a few songs specifically about Southampton:

"The Southampton Tragedy" (concerning the murder of Naomi Kingswell in 1856

"The Knife and Scissor Man" about a Southampton pedler, which mentions lots of place names

The Woolston Ferry about, erm, the Woolston Ferry

I'm too busy and too slow at typing to post them all now, but let me know if you want anything more on any of them.

Ed