  We study area-minimizing hypersurfaces in singular ambient manifolds whose
tangent cones have nonnegative scalar curvature on their regular parts. We
prove that the singular set of the hypersurface has codimension at least 3 in
our settings. We also give an example in which n-dimensional minimizing
hypersurface in an ambient space as given above may have singularities of
dimension $n-3$. This shows that our regularity result is sharp.
