In the search for fast lithium-ion conducting solids for the development of safe rechargeable all-solid-state batteries with high energy density, thiophosphates and related compounds have been demonstrated to be particularly promising both because of their record ionic conductivities and their typically low charge transfer resistances. In this work we explore a wide range of known and predicted thiophosphates with a particular focus on the cubic argyrodite phase with a robust three-dimensional network of ion migration pathways. Structural and hydrolysis stability are calculated employing density functional method in combination with a generally applicable method of predicting the relevant critical reaction. The activation energy for ion migration in these argyrodites is then calculated using the empirical bond valence pathway method developed in our group, while bandgaps of selected argyrodites are calculated as a basis for assessing the electrochemical window. Findings for the lithium compounds are also compared to those of previously known copper argyrodites and hypothetical sodium argyrodites. Therefrom, guidelines for experimental work are derived to yield phases with the optimum balance between chemical stability and ionic conductivity in the search for practical lithium and sodium solid electrolyte materials.