Using the correlative method of unsymmetrized self-consistent field for strongly anharmonic crystals, the thermodynamic stability and the mechanisms of its loss is studied for crystals with various types of the chemical bond. The following interaction potentials are utilized: the Lennard-Jones pairwise potential together with the Axilrod-Teller three-body one for simple van der Waals crystals (solid Ar), the effective interionic pairwise potential proposed by Schiff for a metal (Na) [Phys. Rev. 186, 151 (1969)] and the Girifalco potential for fullerites (C60) [J. Phys. Chem. 96, 858 (1992)]. In the first and third cases, the fcc lattice becomes unstable because the isothermal bulk modulus BT → 0. For fullerites, the shearing coefficient C44 goes to zero as well. Solid Na loses its stability when the other shearing coefficient C11T - C12T becomes zero.