I am looking for a reference to the following result.
Let $f:\mathbb R^m\to\mathbb R$ be a convex function. Then $f$ is differentiable at all points of outside of a countable union of $(m-1)$-rectifiable sets.
$n$-rectifiable set is an image of Lipschitz map from bounded domain in $\mathbb R^n$
I checked Federer's "Geometric Measure Theory", but I might miss the right place.
Extract from the Greg's answer (for those who are lazy to read the paper): In this paper, it is given a complete characterization of subsets of nondifferatiable points of a convex function. Namely, it is proved that $A$ is a such a set if and only if it can be covered by countably many graphs of DC-functions. ("DC-function" = "Difference of Convex functions".)