  In \cite{GGKM-SSS} we examined the relationship between the singular set of a
compact Riemannian orbifold and the spectrum of the Hodge Laplacian on
$p$-forms by computing the heat invariants associated to the $p$-spectrum. We
showed that the heat invariants of the $0$-spectrum together with those of the
$1$-spectrum for the corresponding Hodge Laplacians are sufficient to
distinguish orbifolds from manifolds as long as the singular sets have
codimension $\le 3.$ This is enough to distinguish orbifolds from manifolds for
dimension $\le 3.$ Here we give both positive and negative inverse spectral
results for the individual $p$-spectra considered separately. For example, we
give conditions on the codimension of the singular set which guarantee that the
volume of the singular set is determined, and in many cases we show by
providing counterexamples that the conditions are sharp.
