Let $M$ a connected paracompact differentiable manifold. Let $G$ a connected Lie group. I am interested in the possible "regular" (e.g. smooth) quotients of $M$ by actions of $G$. What topological properties can I expect from them (beside the fibration long exact sequences)? Or maybe they can be organized in some kind of space?

The specific case which interests me is that of smooth quotients by free actions of a non-compact Lie group (principal actions in a broad sense). That being said, results for compact groups (or proper actions) are welcome.

A related question is that of classifying the Lie group actions of $G$ on $M$ up to diffeomorphism: a diffeomorphism conjugating two actions induces a diffeomorphism between the quotient spaces (and identifies the orbit space classifying maps). In the transitive case, the corresponding Lie algebra action is represented by a Maurer-Cartan form, so maybe Lie algebra actions are easier to classify.