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Notes -
It sounds like you know more than me, but I was always astonished that plain old Newtonian mechanics also has singularities and is non-deterministic in certain configurations. It's just so easy for simple-looking sets of equations to unintuitively misbehave.
My PhD dissertation includes a Navier-Stokes problem or two, but getting there took me from typical settings of "best at math in this magnet school" to "second-best at math in this two-person conversation", so don't make any assumptions.
I didn't find Painlevé surprising (you can squeeze a lot out of a situation where you can squeeze point masses arbitrarily deeply into each other's gravitational fields), but I'm annoyed that I never encountered Norton's Dome until after I'd left academia. Those "second-best at math" conversations were humbling but valuable and this is exactly the sort of philosophical+mathematical problem which makes me miss being able to just walk down a hallway and knock on the doors of a bunch of people much smarter than me.
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