The effect of quasiconformal and more general Sobolev maps on the Hausdorff dimensions of individual subsets is well understood. I will present recent results on Hausdorff and Minkowski dimension distortion of generic parallel translates of a fixed linear subspace of Euclidean space under supercritical Sobolev maps. Here genericity is understood in terms of various Hausdorff measures on the orthogonal complement. These results interpolate between the well known ACL property of Sobolev maps and Lusin's condition N, valid for supercritical Sobolev maps. They hold for maps taking values in arbitrary metric spaces, yet are new even for quasiconformal maps of the plane. Time permitting, I will discuss work in progress generalizing these results to nonlinear contexts such as families of linear spaces parameterized by a Grassmannian or non-Riemannian source spaces such as the sub-Riemannian Heisenberg group. This talk is based on joint work with Zoltan Balogh and Roberto Monti.