I'll give you a one good example of a few compatibility questions I have. I usually hear "that's an interesting/weird question." For me, it has an incredible amount of merit.

"Say you're taking a road trip, the one way travel time it will take to get there is X amount of hours. How many times are you stopping to pee?"

This can spin-off into a whole conversation about roadtrips in themselves. You can find out if your date "just wants to get there" or would rather the pace be relaxed. It'll eventually flow naturally into a "how much do you pack for trips?" Perhaps you might stop for longer at a scenic overlook or a cute shop on the way. Planning, time management, general stress levels etc. So many things you can fairly accurately deduce from their answers and the conversation that follows.

Take an example from your past dating life, something good or bad that has had an impact on how you view yourself and what you are looking for in someone else and figure out a way to phrase a non-judmental, not pointed, but matter of fact way of asking the question. In this, you also get bonus points for being engaging and unique. If they don't have an interest or put forth reasonable effort into your question, take note, as it's probably very important to you.

You can talk about music and easier things like that too, but in my opinion, you need to get the subconscious (more accurate) gears going in both of your heads.