  We prove the two theorems of the title, settling two long standing questions
in the local theory of singular minimal hypersurfaces. The sharpness of either
result is with respect to its hypothesis on the size of the allowable singular
sets. The proofs of both theorems rely heavily on the author's recent
regularity and compactness theory for stable minimal hypersurfaces, and on
earlier work of Ilmanen, Simon and Solomon--White.
