  In this paper, the authors consider leaf spaces of singular Riemannian
foliations $\mathcal{F}$ on compact manifolds $M$ and the associated
$\mathcal{F}$-basic spectrum on $M$, $spec_B(M, \mathcal{F}),$ counted with
multiplicities. Recently, a notion of smooth isometry $\varphi:
M_1/\mathcal{F}_1\rightarrow M_2/\mathcal{F}_2$ between the leaf spaces of such
singular Riemannian foliations $(M_1,\mathcal{F}_1)$ and $(M_2,\mathcal{F}_2)$
has appeared in the literature. In this paper, the authors provide an example
to show that the existence a smooth isometry of leaf spaces as above is not
sufficient to guarantee the equality of $spec_B(M_1,\mathcal{F}_1)$ and
$spec_B(M_2,\mathcal{F}_2).$ The authors then prove that if some additional
conditions involving the geometry of the leaves are satisfied, then the
equality of $spec_B(M_1,\mathcal{F}_1)$ and $spec_B(M_2,\mathcal{F}_2)$ is
guaranteed. Consequences and applications to orbifold spectral theory,
isometric group actions, and their reductions are also explored.
