  In N=1 supersymmetric U(N) gauge theory with adjoint matter $\Phi$ and
polynomial tree-level superpotential $W(\Phi)$, the massless fluctuations about
each quantum vacuum are generically described by $U(1)^n$ gauge theory for some
n. However, by tuning the parameters of $W(\Phi)$ to non-generic values, we can
reach singular vacua where additional fields become massless. Using both the
matrix model prescription and the strong-coupling approach, we study in detail
three examples of such singularities: the singularities of the n=1 branch,
intersections of n=1 and n=2 branches, and a class of N=1 Argyres-Douglas
points. In all three examples, we find that the matrix model description of the
low-energy physics breaks down in some way at the singularity.
