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?)