Thank you. That gives me a starting point. Do you have any specific references?

Spinors and spin tensors are the representation theory of SL(2,C), which is the spin group  (universal covering) of SO(1,3). As such they crop up in all relativistic theories because the representation theory of the spin group is the same as the representation theory of the Lie algebra (i.e. the infinitesimal transformations) and is therefore in some sense easier than the tensor representations (although the tensor representations are certainly more intuitive). The application in general relativity is is a purely classical application. IIRC there are even numerical computations of neutron star mergers that are easier to do in spinor tensor form. But the main application of spinors are in quantum field theory (or relativistic quantum mechanics), because that's where bare spinors (rather than spinor representations of tensors) occur.