We consider the spectral problem for the stationary Maxwell's equations under homogeneous boundary conditions in a cavity, and we investigate how the eigenvalues depend on perturbations of the cavity's shape. The problem reduces to the analysis of the curl-curl operator. We study families of diffeomorphic domains and discuss the analytic dependence of the eigenvalues on the domain shape. In particular, we prove Hirakawa’s formula for the shape derivatives and derive a Rellich–Pohozaev-type identity. Furthermore, we address a related shape optimization problem and show that the corresponding Faber–Krahn inequality does not hold. Time permitting, we also consider families of domains undergoing oscillatory boundary perturbations and establish spectral stability results under minimal assumptions on the oscillation strength. In particular, we highlight the role of suitable uniform Gaffney inequalities, which we prove for our purposes.