  In this paper we propose a definition of quasifree Hadamard states for spinor
fields on a curved space-time by specifying the Polarisation Set of the
two-point function. We prove that the thermal equilibrium state on an
ultrastatic space-time is Hadamard. We then construct an adiabatic vacuum state
on a general globally hyperbolic Lorentz manifold using a factorisation of the
spinorial Klein-Gordon operator. This state is pure. In what constitutes the
main part of the paper, we show that it is also Hadamard. As a side result, we
obtain the propagation of singularities of the spinorial Klein-Gordon operator.
Some notation and results are collected in the Appendix.
