  We construct pairs of compact Riemannian orbifolds which are isospectral for
the Laplace operator on functions such that the maximal isotropy order of
singular points in one of the orbifolds is higher than in the other. In one
type of examples, isospectrality arises from a version of the famous Sunada
theorem which also implies isospectrality on $p$-forms; here the orbifolds are
quotients of certain compact normal homogeneous spaces. In another type of
examples, the orbifolds are quotients of Euclidean $\R^3$ and are shown to be
isospectral on functions using dimension formulas for the eigenspaces. In the
latter type of examples the orbifolds are not isospectral on 1-forms. Along the
way we also give several additional examples of isospectral orbifolds which do
not have maximal isotropy groups of different size but other interesting
properties.
