Significance The problem of fitting low-dimensional manifolds to high-dimensional data has been extensively studied from both theoretical and computational perspectives. As datasets get more heterogeneous and complicated, so must the spaces that are used to approximate them. Stratified spaces, built out of manifold pieces coherently glued together, form natural candidates for such geometric models. The key difficulty encountered when fitting stratified spaces to data is that none of the sampled points can be expected to lie exactly on the low-dimensional singular strata. The present work uses local cohomology to overcome this difficulty. Here, we describe an efficient and practical framework for singularity detection from finite samples and demonstrate its ability to detect interfaces in real and simulated data. The quest for low-dimensional models which approximate high-dimensional data is pervasive across the physical, natural, and social sciences. The dominant paradigm underlying most standard modeling techniques assumes that the data are concentrated near a single unknown manifold of relatively small intrinsic dimension. Here, we present a systematic framework for detecting interfaces and related anomalies in data which may fail to satisfy the manifold hypothesis. By computing the local topology of small regions around each data point, we are able to partition a given dataset into disjoint classes, each of which can be individually approximated by a single manifold. Since these manifolds may have different intrinsic dimensions, local topology discovers singular regions in data even when none of the points have been sampled precisely from the singularities. We showcase this method by identifying the intersection of two surfaces in the 24-dimensional space of cyclo-octane conformations and by locating all of the self-intersections of a Henneberg minimal surface immersed in 3-dimensional space. Due to the local nature of the topological computations, the algorithmic burden of performing such data stratification is readily distributable across several processors.