Monte Carlo computer simulation studies have been undertaken for virtually all of the ice phases as well as for liquid water for three of the most popular model potentials; namely SPC/E, TIP4P and TIP5P. Densities have been calculated for specific thermodynamic state points and compared to experimental results. The SPC/E and TIP4P models overestimate the solid densities by about 2%. The TIP5P model overestimates the solid densities by about 5–10%. The structural pair correlation functions between oxygen–oxygen, hydrogen–hydrogen and oxygen–hydrogen atoms were also obtained from the simulations. (These are available as ESI). It has been found that SPC/E and TIP4P structural predictions are rather similar, with the only exception of ice II for which differences are visible between these two models. Predictions from the TIP5P are clearly different from those of the other models, especially for ices I_h and II. For the higher density ices structural differences between the models are rather small. Experimental data would be highly desirable to test the structural predictions of the different models of water. This is especially true for ice II. We have also found that the oxygen–oxygen correlation function of high density amorphous (HDA) water presents the same broad features as those exhibited by ice XII.