We generalize Smirnov's discrete holomorphic observables in the critical Ising model to the case of multiply connected domains. Our observables are spinors, that is, they are multiplicatively multi-valued with monodromy -1. We prove their convergence to conformally covariant scaling limits as the mesh size tends to zero. As applications, we get partial results towards the proof of conformal invariance of the spin correlations, and develop a fairly general theory of scaling limits of multiple Ising interfaces in multiply connected domains.