Azizi has posted a lot of stuff here about "Yo Mamas" and similar kinds of flyting from African-American culture.

There is a surviving tradition like the old Irish bardic one in Kars in north-east Turkey, described in an article in the Garland Encyclopaedia of World Music. Singers get together regularly in cafes to insult each other in rather elaborate improvised songs with saz accompaniment. They have to stick to precise verse forms and metres. Apparently they're hopelessly lost if you ask them to do it without the saz.

There is a different tack on this in some groups in New Guinea, where the insult contests involve embedding your rival's remarks in ever more complicated sentences so as to negate them. (New Guinea languages have extraordinarily complicated syntax even without trying). The winner is the one who comes up with an insult so syntactically intricate that his opponent can't figure out how to negate it.