I am seeking data about war propaganda songs, such as those used in the Israeli/Palestinian conflict, and how they are spread via social media.
I doubt that there's an open data set for this, since it's a new subject and so many parts of it are changing rapidly. Also, it's not terribly clear what kind of data you're looking for, how it would be quantified, etc.
I'd suggest looking for academic research in the field. Some of the researchers may have posted their data, or they may be open to requests for access.
Gilad Lotan has recently published some interesting stuff about the network connections.
Also, I found this one about YouTube: The YouTube Jihadists: A Social Network Analysis of Al-Muhajiroun’s Propaganda Campaign