Recent years have witnessed the development of single-molecule localization microscopy as a generic tool for sampling diverse biologically relevant information at the super-resolution level. While current approaches often rely on the target-specific alteration of the point spread function to encode the multidimensional contents of single fluorophores, the details of the point spread function in an unmodified microscope already contain rich information. Here we introduce a data-driven approach in which artificial neural networks are trained to make a direct link between an experimental point spread function image and its underlying, multidimensional parameters, and compare results with alternative approaches based on maximum likelihood estimation. To demonstrate this concept in real systems, we decipher in fixed cells both the colors and the axial positions of single molecules in regular localization microscopy data.