We prove that area-minimizing hypersurfaces are generically smooth in ambient dimension 11 in the context of the Plateau problem and of area minimization in integral homology. For higher ambient dimensions, n+1 ? 12, we prove in the same two contexts that area-minimizing hypersurfaces have at most an (n ? 10 ? ??)-dimensional singular set after an arbitrarily C?-small perturbation of the Plateau boundary or the ambient Riemannian metric, respectively. This is joint work with O. Chodosh, C. Mantoulidis and Z. Wang.