Philosophaie said:
There are many spaces: Einstein-Cartan Space, Riemann Space, Minkowski Space... Which one does the Earth and the Sun reside in? Which one has Torsion, mass etc. if any?
Russ has basically given the complete answer. I'll toss in my two bits.
You are listing alternative geometries, that go onto space and describe how it acts. Alternative geometries, not different spaces as such.
Maybe a trivial distinction but you probably know the quotes from Einstein where he says points in space have no physical existence, no objective reality.
So the thing to focus on is the geometry. Often it is a dynamic geometry able to interact with matter, behavior governed by a Lagrangian or a differential equation.
So what your question means to me is
which is the best most realistic description of geometry and how it behaves interactively with matter?
I can't tell you any final answer but obviously Minkowski geometry is highly unrealistic. It is only right if there is no matter, and not always even then. It is only approximately right if there is negligible matter in the universe. It does not expand. It is flat. It sucks.
On the other hand (strictly interpreted) Riemannian geometry has the wrong metric signature---which Minkowski at least gets right! So Riemannian is no good.
As Russ hints, all these geometries are human constructs. So the question is which is the most realistic, not which do we live in.
Of the ones you listed I'd go with Einstein Cartan.
But I also like the new version of quantum geometry that came out in 2007. It looks like it might give classical GR in the large distance limit, and also be kind of interesting and weird in the very small distance limit.