My question has two parts, one concerning the state of the art of the subject, and the other the usefulness.
1. State of the art. Can someone provide references reflecting the state of the art in semi-Riemannian geometry with degenerate metric? I am aware of the research of Demir Kupeli, dealing with the case when the signature of the metric is constant. I am mostly interested in the cases when the signature is allowed to vary.
2. Usefulness. Is there a "market" for results extended from the nondegenerate semi-Riemannian geometry to the degenerate one? Would the community of mathematicians and physicists be interested in possible applications, for example to the singularities encountered in General Relativity?
Sorry if these questions seem odd, but I am interested to know whether is worth investing time and resources in doing research in this field. I would like to hear as many opinions as possible.
Update As a matter of fact, I already invested much time and resources, but as I told in an answer to a different question, I would like to make sure I am not reinventing the wheel.
There are also some results concerning cosmological models which start as Riemannian, then, on a hypersurface, the metric becomes Lorentzian.