I can't honestly say I hate any of those genres. But I feel like I have to break it down into which ones I considered choosing.
Pop - Almost chose this one, because I can't stand overplayed radio singles. But then again, the Beatles were pop. Pop is relative to whatever the public wants to hear. Then again, a lot of people who claim to be really into an artist/band tend to like one or two singles, and not bother digging any deeper.
Country - Can't stand modern country (country + pop + rock), but enjoy some of the classics. Anything pre-1980 is pretty decent in my book.
Rap - I don't like the kind of rap that glamorizes degrading women, violence, and living a life of excess, but love clever wordplay, a good flow, and the ability to rhyme well. I saw a few posts along the lines of "I hate rap, but I love some things by Eminem/Beastie Boys/etc". There's a lot more out there than just a massive array of garbage, with occasional artists/groups you can stand (that are still quite popular). It's like saying "I love rock". You won't like ALL rock. I love Queens of the Stone Age but despise Nickelback, even though both are considered rock. Don't judge things you haven't heard before based on a general label.
Also, if I'm jumping headfirst into this, and not pulling any punches, new Eminem sucks.