  We develop explicit techniques to investigate algebraic quasi-hyperbolicity
of singular surfaces through the constraints imposed by symmetric
differentials. We apply these methods to prove that rational curves on Barth's
sextic surface, apart from some well-known ones, must pass through at least
four singularities, and that genus 1 curves must pass through at least two. On
the surface classifying perfect cuboids, our methods show that rational curves,
again apart from some well-known ones, must pass through at least seven
singularities, and that genus 1 curves must pass through at least two.
  We also improve lower bounds on the dimension of the space of symmetric
differentials on surfaces with $A_1$-singularities, and use our work to show
that Barth's decic, Sarti's surface, and the surface parametrizing $3\times 3$
magic squares of squares are all algebraically quasi-hyperbolic.
