Recent simulations of the interaction between planar surfaces and model atomic force microscope (AFM) tips have suggested that there are conditions under which the tip may become unstable and `avalanche' toward the sample surface. Here we investigate via computer simulation the stability of a variety of model AFM tip configurations with respect to the avalanche transition for a number of fcc metals. We perform Monte Carlo simulations at room temperature using the equivalent-crystal theory (ECT) of Smith and Banerjea. Results are compared with recent experimental results as well as with our earlier work on the avalanche of parallel planar surfaces. Our results on a model single-atom tip are in excellent agreement with recent experiments on tunnelling through mechanically controlled break junctions.