  We study local rigidity and multiplicity of constant scalar curvature metrics
in arbitrary products of compact manifolds. Using (equivariant) bifurcation
theory we determine the existence of infinitely many metrics that are
accumulation points of pairwise non homothetic solutions of the Yamabe problem.
Using local rigidity and some compactness results for solutions of the Yamabe
problem, we also exhibit new examples of conformal classes (with positive
Yamabe constant) for which uniqueness holds.
