I know that this sounds really stupid but, when I think of the Minkowski space I cannot imagine a null curve, only null lines. For me, the only possible way to have one is to change the basis of the space for one that is not orthogonal, and that don't make any practical sense for me. And almost the same goes to null surfaces... I just can't think of any other than null planes and null cones. And because of this, I also have doubts of what I think is a null curve and null surface in general relativity.