The whole rift between Neil Young and Lynyrd Skynyrd is overblown. The two bands were friendly, and Ronnie VanZant often went on stage in a Neil Young T-Shirt. Neil Young even said that “Sweet Home Alabama” was a great song and covered it onstage several times. This is more trading friendly barbs than calling someone out.
It’s interesting that someone mentioned New Radical’s calling out of Courtney Love and Marilyn Manson. Didn’t Love and Manson have problems on their '97 tour. I don’t know if it made it into any songs, but they used to insult each other onstage and in interviews quite a bit.
Axl Rose and Vince Neill had a falling out in the early 90’s. Axl somewhat stupidly challenged Vince to a fight on MTV. Vince, who at the time looked like he outweighed Axl by at least 20 pounds immediately took him up on it, and went on MTV himself to call Axl out. Again, I don’t think it got made into a song, but on the video for Crue’s cover of “Anarchy in the U.K.” a fan is filmed holding a sign reading “Vince Neill is going to kick Axl Rose’s ass”
Both John Forgarty and Chrissie Hyndes have implicitly criticized Michael Jackson for selling out to Pepsi.