Is there a way to measure how informal is "What's shaking?"
I think expressions should be better understood than spoken by non-native speakers because it's hard to know exactly when to use them. Unless you have straight contact with the language (being in the English speaking country for example)It's very informal. If you think you have a way to use this, then only use it with people you know very well.
To be sure, "informal" does not mean incorrect. This is a correct expression.
I mean the tone of this expression is very informal.
If you use it, then, of course, you have to be in a situation that is very informal and with people who have a very informal attitude. It has to fit the context or circumstance, too.
This is something that we can hear. However, I'm not sure of how frequently this expression occurs.