  Wahl's local Euler characteristic measures the local contributions of a
singularity to the usual Euler characteristic of a sheaf. Using tools from
toric geometry, we study the local Euler characteristic of sheaves of symmetric
differentials for isolated surface singularities of type $A_n$. We prove an
explicit formula for the local Euler characteristic of the $m$th symmetric
power of the cotangent bundle; this is a quasi-polynomial in $m$ of period
$n+1$. We also express the components of the local Euler characteristic as a
count of lattice points in a non-convex polyhedron, again showing it is a
quasi-polynomial. We apply our computations to obtain new examples of algebraic
quasi-hyperbolic surfaces in $\mathbb{P}^3$ of low degree. We show that an
explicit family of surfaces with many singularities constructed by Labs has no
genus $0$ curves for the members of degree at least $8$ and no curves of genus
$0$ or $1$ for degree at least $10$.
