This thesis gives a classification of infinite staircases for the ellipsoid embedding functions of Hirzebruch surfaces. The ellipsoid embedding function is a generalization of symplectic ball packing problems. For a symplectic manifold, the function gives the smallest amount of which the symplectic form must be scaled in order for a standard ellipsoid of a given eccentricity to embed symplectically into the manifold. Generally, there are only finitely many obstructions other than the volume obstruction relevant to compute the function. If there are infinitely many obstructions, the function is said to have an infinite staircase. This classification problem was studied in a series of five papers written by: Bertozzi-Holm-Maw-McDuff-Mwakyoma-Pires-Weiler, Magill-McDuff, Magill-McDuff-Weiler, Magill, and Magill-Pires-Weiler. The thesis contains two of these papers and includes a summary of the results of the other papers.