Our study addresses the inference of jumps (i.e. sets of discontinuities) within multivariate signals from noisy observations in the non-parametric regression setting. Departing from standard analytical approaches, we propose a new framework, based on geometric control over the set of discontinuities. This allows to consider larger classes of signals, of any dimension, with potentially wild discontinuities (exhibiting, for example, self-intersections and corners). We study a simple estimation procedure relying on histogram differences and show its consistency and near-optimality for the Hausdorff distance over these new classes. Furthermore, exploiting the assumptions on the geometry of jumps, we design procedures to infer consistently the homology groups of the jumps locations and the persistence diagrams from the induced offset filtration.