A surface X is algebraically quasi-hyperbolic if it contains finitely many curves of genus 0 or 1. In 2006, Bogomolov and de Oliveira used asymptotic computations to show that sufficiently nodal surfaces of high degree in projective three-space carry symmetric differentials, and they used this to prove quasi-hyperbolicity of these surfaces. We explain how a granular analysis of their ideas, combined with computational tools and insights, yield explicit results for the existence of symmetric differentials, and we show how these results can be used to give constraints on the locus of rational curves on surfaces like the Barth Decic, Buechi's surface, and certain complete intersections of general type, including the surface parametrizing perfect cuboids, and the surface parametrizing magic squares of squares. This is joint work with Nils Bruin and Jordan Thomas.