The data set "Hydrodynamic Mobility Reversal of Squirmers near Flat and Curved Surfaces" contains the outcome of a theoretical and simulation study into the motion of simple model hydrodynamic swimmers, squirmers, around a spherical obstacle. The main result of this study is that hydrodynamic mobility reversals are possible without the addition of a chemical gradients that help generate the self-propulsion. The robustness of this finding to the curvature of the obstacle with which the squirmer interacts is verified. The results of the analysis are reported on in the associated publication with the same title [10.1039/C9SM00692C]. This data set contains all information necessary to reproduce the graphs in the paper and can be broken down into three categories. The first category, (code.zip) contains all scripts necessary for graph reproduction using post-processed data. This subset comprises predominantly Mathematica notebooks, and latex, python, and gnuplot scripts. The second, (data.tar.gz) contains the means to generate the numerical integration algorithms for our theoretical study, which is based on hydrodynamic Greens functions and the method of reflections; the source code that derives from this analysis; the scripts to generate unprocessed data; and the ones that lead to the post-processed data from which the figures can be generated. This data subset contains python scripts and C++ code. The third category, (dataLB.tar.gz) contains similar information for a complementary numerical study performed using the lattice-Boltzmann method for solving the hydrodynamic equations. This subset contains C++ code, Paraview-compatible vtk files, and python scripts. At the top level, a full description is given in the form of a README.md file that details the manner in which the provided scripts may be used to recreate the figures in the paper.