  We introduce new invariants of a Riemannian singular space, the local Yamabe
and Sobolev constants, and then go on to prove a general version of the Yamabe
theorem under that the global Yamabe invariant of the space is strictly less
than one or the other of these local invariants. This rests on a small number
of structural assumptions about the space and of the behavior of the scalar
curvature function on its smooth locus. The second half of this paper shows how
this result applies in the category of smoothly stratified pseudomanifolds, and
we also prove sharp regularity for the solutions on these spaces. This sharpens
and generalizes the results of Akutagawa and Botvinnik \cite{AB} on the Yamabe
problem on spaces with isolated conic singularities.
