Many-body perturbation theory in the $GW$ approximation is a useful method for describing electronic properties associated with charged excitations. A hierarchy of $GW$ methods exists, starting from non-self-consistent ${G}_{0}$${W}_{0}$, through partial self-consistency in the eigenvalues and in the Green's function (sc$G{W}_{0}$), to fully self-consistent $GW$ (sc$GW$). Here, we assess the performance of these methods for benzene, pyridine, and the diazines. The quasiparticle spectra are compared to photoemission spectroscopy (PES) experiments with respect to all measured particle removal energies and the ordering of the frontier orbitals. We find that the accuracy of the calculated spectra does not match the expectations based on their level of self-consistency. In particular, for certain starting points ${G}_{0}$${W}_{0}$ and sc$G{W}_{0}$ provide spectra in better agreement with the PES than sc$GW$.