  For distributions, we build a theory of higher order pointwise
differentiability comprising, for order zero, {\L}ojasiewicz's notion of point
value. Results include Borel regularity of differentials, higher order
rectifiability of the associated jets, a Rademacher-Stepanov type
differentiability theorem, and a Lusin type approximation. A substantial part
of this development is new also for zeroth order. Moreover, we establish a
Poincar\'e inequality involving the natural norms of negative order of
differentiability. As a corollary, we characterise pointwise differentiability
in terms of point values of distributional partial derivatives.
