  We describe the loci of non-rationally smooth (nrs) points and of singular
points for any non-spiral Schubert variety of $\tilde{A}_2$ in terms of the
geometry of the (affine) Weyl group action on the plane $\mathbb{R}^2$.
Together with the results of Graham and Li for spiral elements, this allows us
to explicitly identify the maximal singular and nrs points in any Schubert
variety of type $\tilde{A}_2$. Comparable results are not known for any other
infinite-dimensional Kac-Moody flag variety (except for type $\tilde{A}_1$,
where every Schubert variety is rationally smooth). As a consequence, we deduce
that if $x$ is a point in a non-spiral Schubert variety $X_w$, then $x$ is nrs
in $X_w$ if and only if there are more than $\dim X_w$ curves in $X_w$ through
$x$ which are stable under the action of a maximal torus, as is true for
Schubert varieties in (finite) type $A$. Combined with the work of Graham and
Li for spiral Schubert varieties, this implies the Lookup Conjecture for
$\tilde{A}_2$.
