Previous studies have shown that most random amino acid substitutions destabilize protein folding (i.e. increase the folding free energy). No analogous studies have been carried out for protein-protein binding. Here we use a structure-based model of the major coat protein in a simple virus, bacteriophage φX174, to estimate the free energy of folding of a single coat protein and binding of five coat proteins within a pentameric unit. We confirm and extend previous work in finding that most accessible substitutions destabilize both protein folding and protein-protein binding. We compare the pool of accessible substitutions with those observed among the φX174-like wild phage and in experimental evolution with φX174. We find that observed substitutions have smaller effects on stability than expected by chance. An analysis of adaptations at high temperatures suggests that selection favors either substitutions with no effect on stability or those that simultaneously stabilize protein folding and slightly destabilize protein binding. We speculate that these mutations might involve adjusting the rate of capsid assembly. At normal laboratory temperature there is little evidence of directional selection. Finally, we show that cumulative changes in stability are highly variable; sometimes they are well beyond the bounds of single substitution changes and sometimes they are not. The variation leads us to conclude that phenotype selection acts on more than just stability. Instances of larger cumulative stability change (never via a single substitution despite their availability) lead us to conclude that selection views stability at a local, not a global, level.