  In this paper, we study noncommutative surface singularities arising from
orders. The singularities we study are mild in the sense that they have finite
representation type or, equivalently, are log terminal in the sense of the Mori
minimal model program for orders [CI05]. These were classified independently by
Artin (in terms of ramification data) and Reiten-Van den Bergh (in terms of
their AR-quivers). The first main goal of this paper is to connect these two
classifications, by going through the finite subgroups $G \subset
\mathrm{GL}_2$, explicitly computing $H^2(G,k^*)$, and then matching these up
with Artin's list of ramification data and Reiten-Van den Bergh's AR-quivers.
This provides a semi-independent proof of their classifications and extends the
study of canonical orders in [CHI09] to the case of log terminal orders. A
secondary goal of this paper is to study noncommutative analogues of plane
curves which arise as follows. Let $B = k_{\zeta} [[ x,y ]]$ be the skew power
series ring where $\zeta$ is a root of unity, or more generally a terminal
order over a complete local ring. We consider rings of the form $A = B/(f)$
where $f \in Z(B)$ which we interpret to be the ring of functions on a
noncommutative plane curve. We classify those noncommutative plane curves which
are of finite representation type and compute their AR-quivers.
