We develop explicit techniques to investigate algebraic quasihyperbolicity 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 A1-singularities, and use our work to show that Barth’s decic, Sarti’s surface, and the surface parametrizing 3×3 magic squares of squares are all algebraically quasihyperbolic.