  We study the singular values of the Dirac operator in dense QCD-like theories
at zero temperature. The Dirac singular values are real and nonnegative at any
nonzero quark density. The scale of their spectrum is set by the diquark
condensate, in contrast to the complex Dirac eigenvalues whose scale is set by
the chiral condensate at low density and by the BCS gap at high density. We
identify three different low-energy effective theories with diquark sources
applicable at low, intermediate, and high density, together with their
overlapping domains of validity. We derive a number of exact formulas for the
Dirac singular values, including Banks-Casher-type relations for the diquark
condensate, Smilga-Stern-type relations for the slope of the singular value
density, and Leutwyler-Smilga-type sum rules for the inverse singular values.
We construct random matrix theories and determine the form of the microscopic
spectral correlation functions of the singular values for all nonzero quark
densities. We also derive a rigorous index theorem for non-Hermitian Dirac
operators. Our results can in principle be tested in lattice simulations.
