  We develop a general regulated volume expansion for the volume of a manifold
with boundary whose measure is suitably singular along a separating
hypersurface. The expansion is shown to have a regulator independent anomaly
term and a renormalized volume term given by the primitive of an associated
anomaly operator. These results apply to a wide range of structures. We detail
applications in the setting of measures derived from a conformally singular
metric. In particular, we show that the anomaly generates invariant
(Q-curvature, transgression)-type pairs for hypersurfaces with boundary. For
the special case of anomalies coming from the volume enclosed by a minimal
hypersurface ending on the boundary of a Poincare--Einstein structure, this
result recovers Branson's Q-curvature and corresponding transgression. When the
singular metric solves a boundary version of the constant scalar curvature
Yamabe problem, the anomaly gives generalized Willmore energy functionals for
hypersurfaces with boundary. Our approach yields computational algorithms for
all the above quantities, and we give explicit results for surfaces embedded in
3-manifolds.
