  We study the question of existence of a Riemannian metric of positive scalar
curvature metric on manifolds with the Sullivan-Baas singularities. The
manifolds we consider are Spin and simply connected. We prove an analogue of
the Gromov-Lawson Conjecture for such manifolds in the case of particular type
of singularities. We give an affirmative answer when such manifolds with
singularities accept a metric of positive scalar curvature in terms of the
index of the Dirac operator valued in the corresponding "K-theories with
singularities". The key ideas are based on the construction due to Stolz, some
stable homotopy theory, and the index theory for the Dirac operator applied to
the manifolds with singularities. As a side-product we compute homotopy types
of the corresponding classifying spectra.
