In my master's thesis I developed the theory of G-structures on effective orbifolds.
Orbifolds are well-behaved singular spaces on which we can use differentiable calculus.
The theory of G-structures allow us to pass from questions about geometric structures on manifolds/orbifolds to questions about G-equivariant differentiable forms on a principal bundle: the frame bundle. In this way we can study invariants of geometric structures by using Cartan calculus on G-equivariant forms.
The main advantage of using this theory for orbifolds is that the frame orbibundle of an effective orbifold is a manifold and then we can study geometric structures on singular objects, orbifolds, by studying the G-equivariant geometry on the manifold of frames.
Here is a pdf of my master's thesis.
A word of warning: there are a few non-essential mistakes on the thesis so feel free to ask me about more details.