Both are OK, but I would generally stick with the singular "kind" when I ask this kind of question.
You raise an interesting point, but it's a bit of a tangent from your question ...
If you ask "what kinds" then you're assuming that I like more than one kind. Take another one:
- What sport do you like?
- What sports do you like?
- Do you prefer football or basketball?
- Do you like (any) sports?
Each of the first three questions makes an assumption about me, that I like sport. (I don't. I hate sports!)
While it's generally not a big problem to assume that most people like some kind of music or sport, I have noticed that my Chinese students seem to ask this kind of question more often than people from Western countries. I think there is some cultural difference here. It's just a thought, not really a big deal, but keep it in your mind as you learn more language. Eventually, language study becomes cultural comparison. We think in different ways, even though we use the same words. If you can bridge that gap, communication gets a lot easier.