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-εₙ dimensional singular set after an arbitrarily C^-small perturbation of the Plateau boundary or the ambient Riemannian metric, respectively.
Chodosh et al. (Sun,) studied this question.