See the paper: MR2598628 Reviewed Steinbauer, Roland: A note on distributional semi-Riemannian geometry. Novi Sad J. Math. 38 (2008), no. 3, 189–199.
See also the book: MR1883263 Reviewed Grosser, Michael; Kunzinger, Michael; Oberguggenberger, Michael; Steinbauer, Roland Geometric theory of generalized functions with applications to general relativity. Mathematics and its Applications, 537. Kluwer Academic Publishers, Dordrecht, 2001. xvi+505 pp.
There singular semi-Riemannian metrics are studied in the sense of distributions. But there is the need to multiply distributions in order to compute curvature and check Einstein's equation. So this uses an extension of distributions where you can multiply, but loose some properties.
Edit: You find many papers in this directions by looking here