Could spacetimes have singular manifolds? Let's take a spacetime as a pair $(M,g)$ where $M$ is the manifold and $g$ the metric.
I've seen that there exist a generalization of manifolds. This generalization consist in accept singularities in the manifold ( http://www.map.mpim-bonn.mpg.de/Manifolds_with_singularities#:~:text=Manifolds%20with%20singularities%20are%20geometric,are%20understood%20to%20be%20smooth. here we have a more formal definition and some works on this subject).
Some spacetimes have singularities defined by the incompleteness of geodesics, but these singularities are associated with the metric and not with the manifold in the sense that manifolds are (let's say) "not singular". Could a singular spacetime $(M',g)$ with $M'$ a manifold with singularities exist or is it senseless? (If it is possible, should $g$ be singular as well or is it possible a pair $(M',g)$ with $g$ non-singular?)
 A: Actually, in string theory they consider stringy sigma models with target spaces an orbifold. Orbifolds are spaces with a well understood form of singularity.
Another example of a singular space occurs in non-commutative geometry, specifically in the Barrett-Connes-Lott-Chamseddine model of the classical Standard Model, ie before quantisation, reproducing it in quite fine detail, including neutrino mixing.
Here, a spectral triple encodes a product of the geometric 4d Minkowski space with a non-geometric space whose classical dimension is that of a point, but which non-classically is 6d.
In fact, their spectral action is the non-commutative version of the Kaluza-Klien mechanism. So in fact, they are postulating a higher gravitational field that via their action reproduces the classical fields of the standard model.
A: The link gives examples such as the union of two spheres, and in an example like this, any Riemannian metric defined on the spheres gets extended to one that is singular where the spheres intersect. This seems to me to be a setup that is guaranteed not to be of interest in classical relativity. The singularity guarantees that observers in a particular region can never do observations or make predictions about the regions that lie beyond the singularty.
