  We obtain a criterion for an analytic subset of a Euclidean space to contain
points of differentiability of a typical Lipschitz function, namely, that it
cannot be covered by countably many sets, each of which is closed and purely
unrectifiable (has zero length intersection with every $C^1$ curve).
Surprisingly, we establish that any set failing this criterion witnesses the
opposite extreme of typical behaviour: In any such coverable set a typical
Lipschitz function is everywhere severely non-differentiable.
