Advances in protein structure determination and modeling allow us to study the structural context of human genetic variants on an unprecedented scale. Here, we analyze millions of cancer-associated missense mutations based on their structural locations and predicted perturbative effects. By considering the collective properties of mutations at the level of individual proteins, we identify distinct patterns associated with tumor suppressors and oncogenes. Tumor suppressors are enriched in structurally damaging mutations, consistent with loss-of-function mechanisms, while oncogene mutations tend to be structurally mild, reflecting selection for gain-of-function driver mutations and against loss-of-function mutations. Although oncogenes are difficult to distinguish from genes with no role in cancer using only structural damage, we find that the three-dimensional clustering of mutations is highly predictive. These observations allow us to identify candidate driver genes and speculate about their molecular roles, which we expect will have general utility in the analysis of cancer sequencing data.