  We establish the existence of infinitely many complete metrics with constant
scalar curvature on prescribed conformal classes on certain noncompact product
manifolds. These include products of closed manifolds with constant positive
scalar curvature and simply-connected symmetric spaces of noncompact or
Euclidean type; in particular, $\mathbb S^m \times\mathbb R^d$, $m\geq2$,
$d\geq1$, and $\mathbb S^m\times\mathbb H^d$, $2\leq d<m$. As a consequence, we
obtain infinitely many periodic solutions to the singular Yamabe problem on
$\mathbb S^m\setminus\mathbb S^k$, for all $0\leq k<(m-2)/2$, the maximal range
where nonuniqueness is possible. We also show that all Bieberbach groups in
$Iso(\mathbb R^d)$ are periods of bifurcating branches of solutions to the
Yamabe problem on $\mathbb S^m\times\mathbb R^d$, $m\geq2$, $d\geq1$.
