Suppose $M$ is a smooth manifold and a compact Lie group $G$ acts freely on $M$, then it is well known that $M/G$ has a manifold structure.

Are there any results for the general case? (a) If the action is not free, what can we say about the local structure of the quotient? Can we define a stratification on the space? How regular are the strata? (b) Moreover, what can we say if $M$ and $G$ are Hilbert manifolds?

Thanks a lot!