  We examine 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 show 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 with
singularities from manifolds as long as the singular sets have codimension $\le
3.$ This is enough to distinguish orbifolds from manifolds for dimension $\le
3.$
