I can't speak for everyone, but I do listen to a little bit of everything. I definitely have preferences for certain things though, mostly rock, hip-hop, but I definitely dabble in other genres. Variety is key.
I say mostly rock and metal. Then I usually specify genres and/or bands. If further inquiries are made, I go into more different genres that don't interest me as much. It usually doesn't get to this point, though.
I say everything most of the time. Its easier. Most people who ask that question are more into indie bands or stuff that is "in" now. I dont have anything against that kind of music, but I just dont come across it when I browse music.
Most definitions of "everything" are pretty narrow. It's generally said by those who are too ignorant/lazy to understand that there is music outside of mainstream exposure. To me it means finding appeal in music without regard to time, language, culture/nationality, style/genre, popularity etc. (i.e. everything), which is why I never take anyone who says they listen to "everything" seriously. I never use that description for what I like; there's plenty of music out there that made me want to rip out my inner ear, though it's still easier for me to say what music I don't like listening to.