When a sphere, suspended in an otherwise quiescent fluid, is rotated, the resulting flow exhibits a wide range of fundamental fluid physics, ranging from the collision of boundary layers through to the development of radial jets and subsequent absolute instabilities in the spatially and temporally developing flow. In this way, such a flow provides a paradigm for the study of instabilities in temporally and spatially developing boundary layers. There is a body of experimental work on this problem Y. Kohama and R. Kobayashi, J. Fluid Mech. 137, 153 (1983); T. Hada and A. Ito, Trans. Vis. Soc. Jpn. 23, 231 (2003) that suggests that the flow within the sphere's boundary layer can be unstable to spiral vortices for moderate rotation rates; however, other experiments F. P. Bowden and R. G. Lord, Proc. R. Soc. London A 271, 143 (1963); F. Kreith et al., Int. J. Heat Mass Transfer 6, 881 (1963); S. A. W. Calabretto et al. (unpublished) demonstrate that the flow over the sphere remains laminar until much higher rotation rates. We consider this problem from a computational perspective, seeking to understand how, when, and where such instabilities may develop. Our results show good agreement with the experiments of Kreith et al. and Calabretto et al. in that the flow over much of the boundary layer remains undisturbed and, additionally, the spiral vortex instabilities that develop in the flow are convective in nature.