I thought light like singularity is where the geodesics end on a lightlike hypersurface and can't be extended anymore. I guess its different than light cone singularity. Lot's of places have mention of it, but I wanted to know its definition for sure.
You may have encountered it in a different context, but I recognize it from the topic of singularities of correlation functions in quantum field theory. In massless field theories such correlation fucntions becomes singular for points which are lightlike separated, and the structure of such singularities is determined by good physical principles such as locality and unitarity.
Then again, I may be completely off the mark. In any event, except for the linguistic similarity, I don't think it has to do with null singularities in spacetime.