  Suppose that $f$ defines a singular, complex affine hypersurface. If the
critical locus of $f$ is one-dimensional at the origin, we obtain new general
bounds on the ranks of the homology groups of the Milnor fiber, $F_{f, \mathbf
0}$, of $f$ at the origin, with either integral or $\mathbb Z/p\mathbb Z$
coefficients. If the critical locus of $f$ has arbitrary dimension, we show
that the smallest possibly non-zero reduced Betti number of $F_{f, \mathbf 0}$
completely determines if $f$ defines a family of isolated singularities, over a
smooth base, with constant Milnor number. This result has a nice interpretation
in terms of the structure of the vanishing cycles as an object in the perverse
category.
