I have explored this a bit in therapy recently.
I've found that it definitely has some obsessive/compulsive components to it and so I have been making myself aware of when it happens and what happens when a song is stuck in my head. (Lately it's been Joni Mitchell's songs from Ladies of the Canyon CD).
What I found is that having the songs, or bits of the songs, playing over and over in my head were a way of avoiding thinking about other things in my life that needed my attention. When I can shut it off, I feel calmer and clearer-headed, more present / in the moment.

I now try to keep it from happening and it helps me stay grounded. If I'm actively listening to it, that's one thing, but if I'm anxiously listening to it in my head then it's time to pay attention to what is going on.