The open singularity strata (folds, cusps, etc.) of a generic smooth map f: N->M are well known to be smooth manifolds. Their closures in N need not be locally Euclidean in general, but often admit canonical resolutions by closed smooth manifolds. A general approach to finding such resolutions (possibly in all cases) will be presented, based on viewing singularities as limit points of multiple point loci of f. Technically, it amounts to extending the multijet transversality theorem over a new polynomial compactification of configuration spaces. Examples and partial results will be discussed.