In the vast majority of situations, we will just say "not much," and then proceed to tell you what's going on. For example:
- What's up?
- Not much, you? (or "what's up with you?")
or
- What's up?
- Not much, I've just got to drop off the dog at the groomer before I pick up the kids from school and take them to band practice. And later on Sherri and I are going to a show. You?
So you see that even though I started with "not much," it's not true. I am speaking about American + Canadian English, in this case.
However, if the person asking "What's up?" says it with concern in their voice, with a worried kind of intonation, we would answer differently:
- What's up? (I think you look sad)
- Oh, nothing. I'm just tired.
or
- What's up? (I think you look sad)
- To be honest I've had better days. My dog is missing.
Context is everything, but similarly to how English folks use "Alright?" without really thinking about the meaning, and then carry on with the actual content of the conversation afterwards, Americans will use "what's up" almost like "hello." (almost!)