Nonmagnetic impurities are known as strong pair breakers in superconductors with pure d-wave pairing symmetry. Here we discuss d-wave states under the combined influence of impurities and competing instabilities, such as pairing in a secondary channel as well as lattice symmetry breaking. Using the self-consistent T-matrix formalism, we show that disorder can strongly modify the competition between different pairing states. For a d-wave superconductor in the presence of a subdominant local attraction, Anderson's theorem implies that disorder always generates an s-wave component in the gap at sufficiently low temperature, even if a pure d _x2-y2 order parameter characterizes the clean system. In contrast, disorder is always detrimental to an additional d _xy component. This qualitative difference suggests that disorder can be used to discriminate among different mixed-gap structures in high-temperature superconductors. We also investigate superconducting phases with lattice symmetry breaking in the form of bond order and show that the addition of impurities quickly leads to the restoration of translation invariance. Our results highlight the importance of controlling disorder for the observation of competing order parameters in cuprates.