Pattern-formation processes at solid/solid interfaces are investigated by Monte Carlo simulations using an appropriate two-dimensional model system AX/BX with an initially planar, coherent interface. The motion of cations A and B occurs via vacancies in the regular cation sublattice, and the jump frequencies of both A and B are described by a simple Boltzmann-ansatz. Therefore, the jump frequency is a function of temperature and the nearest neighbourhood of each cation, whose influence is determined by repulsive pairwise interaction energies ε_AA, ε_AB and ε_BB. Using appropriate boundary conditions, a directed vacancy flux and, therefore, the growth of one or both phases is caused. In this way an external force, e.g. an external electric field, in a real experiment is simulated. Below a critical temperature T_C (limited miscibility between AX and BX), the phase boundary roughens but remains morphologically stable if the less mobile phase is the growing phase. In comparison, a critical parameter Δε_C= ε_AA-ε_BB is observed, above which the phase boundary becomes morphologically unstable if the more mobile phase is the growing one. In this case Δε_C is dependent on temperature, the boundary conditions and the external driving force. With increasing Δε a transition from finger-like to branched structures is observed. In exceptional cases the latter can be described as fractals. Similar results are obtained at temperatures above T_C (complete miscibility of AX and BX) where we find instabilities of diffusion fronts.