  We establish a general structure theorem for the singular part of ${\mathscr
A}$-free Radon measures, where ${\mathscr A}$ is a linear PDE operator. By
applying the theorem to suitably chosen differential operators ${\mathscr A}$,
we obtain a simple proof of Alberti's rank-one theorem and, for the first time,
its extensions to functions of bounded deformation (BD). We also prove a
structure theorem for the singular part of a finite family of normal currents.
The latter result implies that the Rademacher theorem on the differentiability
of Lipschitz functions can hold only for absolutely continuous measures and
that every top-dimensional Ambrosio--Kirchheim metric current in $\mathbb R^d$
is a Federer-Fleming flat chain.
