We report a lattice-dynamics study of relative stability of various phases of natural silicates MSiO₄ (M =Zr, Hf, Th, and U as a function of pressure (P) and temperature (T), which is important in the context of their use in nuclear waste storage media. Extending our previous work on ZrSiO₄, the Gibbs free energy has been calculated using a transferable interatomic potential in various phases over a range of P and T. Due to an interesting interplay between the vibrational entropy and atomic packing, the zircon body-centered tetragonal, I4₁/amd), scheelite body-centered tetragonal, I4₁/a), and huttonite (monoclinic, P2₁/n)  phases occur at different P and T. It is shown that, for ThSiO₄ at high P, the huttonite and scheelite phases are favored at high and low T, respectively. However, for both USiO₄ and HfSiO₄ the huttonite phase is dynamically unstable and the scheelite phase is stable as the high pressure phase. In fact, the calculations reveal that the stability of the huttonite phase is determined by the ionic size of the M atom; this phase is unstable for the silicate with the smaller Hf and U ions, and stable with the larger Th ion. The calculated phase diagrams are in fair agreement with the reported experimental observations. The calculated structures, phonon spectra, and various thermodynamic properties also fairly well reproduce the available experimental data. The low-energy phonons in the huttonite phase that contribute to its large vibrational entropy are found to involve librational motion of the silicate tetrahedral units.