We consider the ground state and the low-lying excitations of dipolar Bose-Einstein condensates in a bubble trap, i.e., a shell-shaped spherically symmetric confining potential. By means of an appropriate Gaussian ansatz, we determine the ground-state properties in the case where the particles interact by means of both the isotropic and short-range contact and the anisotropic and long-range dipole-dipole potential in the thin-shell limit. Moreover, with the ground state at hand, we employ the sum-rule approach to study the monopole, the two-, the three-dimensional quadrupole as well as the dipole modes. We find situations in which neither the virial nor Kohn's theorem can be applied. On top of that, we demonstrate the existence of anisotropic particle density profiles, which are absent in the case with repulsive contact interaction only. These significant deviations from what one would typically expect are then traced back to both the anisotropic nature of the dipolar interaction and the novel topology introduced by the bubble trap.