The paper discusses the phase-space structure of six variants of a simple mechanical system that differ in the applied friction model. It is shown that many properties of the Coulomb and the Stribeck models, such as the number of equilibria and their stability, are inherited by the Dahl and the LuGre dynamic friction models, respectively. Exploiting similar relationships, a Coulomb-based and a Stribeck-based version of the Generalized Maxwell-Slip model are also introduced. The detailed analysis of these models reveals a surprisingly rich variety of equilibrium types and bifurcations. Moreover, it is highlighted that the most frequently used values of the Stribeck exponent may lead to atypical results such that even a small deviation from these values changes the bifurcation scenario.