The set of umbilics of infinite curvature of a "typical" convex body in $\mathbb{R}^n$ has measure zero, but is dense and uncountable.
Here "typical" is in the sense of Baire category: the subset of convex bodies with this property is "comeager."
Schneider, Rolf. "Curvatures of typical convex bodies—The complete picture." Proc. Amer. Math. Soc., (PDF download link).
There are several properties of typical convex bodies that are somewhat counterintuitive.