  Let $M_\Sigma$ be an $n$-dimensional Thom-Mather stratified space of depth
$1$. We denote by $\beta M$ the singular locus and by $L$ the associated link.
In this paper we study the problem of when such a space can be endowed with a
wedge metric of positive scalar curvature. We relate this problem to recent
work on index theory on stratified spaces, giving first an obstruction to the
existence of such a metric in terms of a wedge $\alpha$-class $\alpha_w
(M_\Sigma)\in KO_n$. In order to establish a sufficient condition we need to
assume additional structure: we assume that the link of $M_\Sigma$ is a
homogeneous space of positive scalar curvature, $L=G/K$, where the semisimple
compact Lie group $G$ acts transitively on $L$ by isometries. Examples of such
manifolds include compact semisimple Lie groups and Riemannian symmetric spaces
of compact type. Under these assumptions, when $M_\Sigma$ and $\beta M$ are
spin, we reinterpret our obstruction in terms of two $\alpha$-classes
associated to the resolution of $M_\Sigma$, $M$, and to the singular locus
$\beta M$. Finally, when $M_\Sigma$, $\beta M$, $L$, and $G$ are simply
connected and $\dim M$ is big enough, and when some other conditions on $L$
(satisfied in a large number of cases) hold, we establish the main result of
this article, showing that the vanishing of these two $\alpha$-classes is also
sufficient for the existence of a well-adapted wedge metric of positive scalar
curvature.
