Oh man, I really can't answer that. First of all, I don't have a favorite band (at all) so it depends entirely on the band and the song. I'm a singer myself, so obviously I pay a lot of attention to that, but I was also the primary music arranger in my a cappella group, so I automatically break down music I hear into parts.
It tends to be whatever stands out the most to me in a song. For example, in the Foo Fighters song Pretender, the bass line stands out to me the most. Basically I tend to first hear the ear markers I would use for an a cappella arrangement. Often it's the bass, because getting that down sets up an excellent base (no pun intended) for an arrangement, often it's a hook or recognizable riff (again, harkening back to arranging - if you get the signature riff or hook down and you get the bass down, you can sort of play with the other parts) or if the most recognizable part is the guitar pattern, I'll hear that first. Ill even do drums, since for one year I did a lot of vocal percussion for songs that needed it.
Short story long (sorry) I basically tend to listen to every song as if it's an a cappella song and sing whichever part pops out at me or, if the vocal line is cool, I go for that.
Fincham: Where are you going?
Me: I have no idea
Fincham: I meant where are you sitting. This wasn't an existential question.