There are two types there for me: the tunes that stick because I'm listening to them often and love them, and the ones that stick even if I don't listen to them anymore.
First case, 'worst' one is Knoxville, Summer of 1915 op. 24 (Barber). I love that piece, the dusty atmosphere, the incredible universality that emanates from it, even if the title and the text itself indicate an extremely specific setting. (this piece is actually a (unlikely!) big inspiration for my comic).
But it comes so often in my head that it's almost worrying. I should learn to actually sing it, it could make it easier to deal with it, but I would need to adapt it a bit to be able to sing it, and I simply don't take the time.
Very different, the case where I don't even listen to the song but it's sticking for years: Damned Ladies by Rufus Wainwright is the clearer case.
I loved Wainwright's first album back in 1998 (actually saw him live 3 times back then) but this is really something I grew out of.
But that song stayed in my head, especially the part that says "There is a knock at the door/tell me it's not Mimi again/or is it Gilda's waiting passion to be stabbed and killed again?" Funniest thing is that he and I have pretty much opposite tastes in opera, and I have a strong dislike for most of the operas (at least the Italian ones) cited in that song, so I say the fact it's so strongly stuck in my head is the vengeance of Italian opera 
