Spin structures and spectra of Z₂^k-manifolds

We give necessary and sufficient conditions for the existence of pin+, pin- and spin structures on Riemannian manifolds with holonomy group $Z_2^k$. For any n>3 (resp. n>5) we give examples of pairs of compact manifolds (resp. compact orientable manifolds) M_1, M_2, non homeomorphic to each other, that are Laplace isospectral on functions and on p-forms for any p and such that M_1 admits a pin+, or pin-, (resp. spin) structure whereas M_2 does not.