  Non-magnetic 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_{x^2-y^2} 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.
