We report unphysical irregularities and discontinuities in some key experimentally measurable quantities computed within the GW approximation of many-body perturbation theory applied to molecular systems. In particular, we show that the solution obtained with partially self-consistent GW schemes depends on the algorithm one uses to self-consistently solve the quasiparticle (QP) equation. The main observation of the present study is that each branch of the self-energy is associated with a distinct QP solution and that each switch between solutions implies a significant discontinuity in the quasiparticle energy as a function of the internuclear distance. Moreover, we clearly observe "ripple" effects, i.e., when a discontinuity in one of the QP energies induces (smaller) discontinuities in the other QP energies. Going from one branch to another implies a transfer of weight between two solutions of the QP equation. The cases of occupied, virtual, and frontier orbitals are separately discussed on distinct diatomics. In particular, we show that multisolution behavior in frontier orbitals is more likely if the HOMO-LUMO gap is small.