If I understand correctly, we roughly define the singular locus $\Sigma_O$ of an orbifold $O$ to be the set of points where the orbifold fails to be a manifold. In particular, if $x \in O$ has a neighbourhood $U \subseteq X_O$ which is homeomorphic to $\bar{U}/\Gamma_i$ for $\bar{U} \subseteq \mathbb{R}^n$, then we can associate to $x$ the isotropy group $I_{\bar{x}}$ of any $\bar{x} \in \bar{U}$ projecting to $x$, which is well defined up to isomorphism. We then have $$ \Sigma_0 = \{ x \in X_O : I_x \ne \emptyset \}.$$ Now, for two-dimensional orbifolds $O$, it turns out that every singular point $x \in \Sigma_O$ has a neighbourhood modelled on either $\mathbb{R}^2/Z_2$, $\mathbb{R}^2/Z_n$, or $\mathbb{R}^2/D_n$. We call such points mirror points, elliptic points, or corner reflectors, respectively.
My question is really one of visualisation: How should I visualise these different classes? I know for example that elliptic points are exemplified by taking the quotient of $\mathbb{R}^2$ by $C_n$ acting by rotations about the origin, giving you an infinite cone with an elliptic point at the apex. But what are some nice, easily visualised examples of corner reflectors and mirrors? Thanks.