Let $\Sigma$ be a two-dimensional connected smooth manifold without boundary. (Feel free to assume it is compact and orientable.)

Let $\mathcal{X}(\Sigma)$ denote the smooth vector fields on $\Sigma$ and let $\operatorname{Diff}(\Sigma)$ denote the group of diffeomorphisms of $\Sigma$. Clearly $\operatorname{Diff}(\Sigma)$ acts on $\mathcal{X}(\Sigma)$.

I would expect naively that the space of orbits $\mathcal{M}:=\mathcal{X}(\Sigma)/\operatorname{Diff}(\Sigma)$ would be finite-dimensional. Is this actually the case?

Question

What can one say about $\mathcal{M}$ in general?

Any references where I could read about this question would be appreciated.