  Let $k$ be a field, $X$ a variety with tame quotient singularities and
$\tilde{X}\to X$ a resolution of singularities. Any smooth rational point $x\in
X(k)$ lifts to $\tilde{X}$ by the Lang-Nishimura theorem, but if $x$ is
singular this might be false.
  For certain types of singularities the rational point is guaranteed to lift,
though; these are called singularities of type $\mathrm{R}$. This concept has
applications in the study of the fields of moduli of varieties and yields an
enhanced version of the Lang-Nishimura theorem where the smoothness assumption
is relaxed.
  We classify completely the tame quotient singularities of type $\mathrm{R}$
in dimension $2$; in particular, we show that every non-cyclic tame quotient
singularity in dimension $2$ is of type $\mathrm{R}$, and most cyclic
singularities are of type $\mathrm{R}$ too.
