Let $\mathbb{H}$ be an upper half plane (this is a Riemann surface), then $PSL(2,\mathbb{Z})$ acts on $\mathbb{H}$ and it is well-know that $ \mathbb{H}/PSL(2,\mathbb{Z})\cong \mathbb{C} $ is again a Riemann surface. There are however three fixed points on $\mathbb{H}$, namely $ e^{\frac{2\pi i}{6}}, \ \ \ i, \ \ \ e^{\frac{2\pi i}{3}}. $ I hence think the image of these points should be considered as orbifold point of $\mathbb{H}/PSL(2,\mathbb{Z})$. I know there is a map $z\mapsto z^n$ which gives a new local parameter at each orbifold point, but is it really natural? It seems to me that this new chart is not really natural as it is not conformal at the orbifold point (it maps an angle $\theta$ to $n\theta$, right?)
More generally if a finite group $G$ acts on a Riemann surface $S$, we can ask a similar question about the quotient $S/G$. Should one think of $S/G$ as a smooth Riemann surface or an orbifold?