The future detection of gravitational waves (GWs) from a galactic core-collapse supernova will provide information on the physics inside protoneutron stars (PNS). In this work, we apply three different classification methods for the PNS non-radial oscillation modes: Cowling classification, Generalized Cowling Nomenclature (GCN), and a Classification Based on Modal Properties (CBMP). Using PNS models from 3D simulations of core-collapse supernovae, we find that in the early stages of the PNS evolution, typically before 0.4 seconds after the bounce, the Cowling classification is inconsistent, but the GCN and the CBMP provide complementary information that helps to understand the evolution of the modes. In the GCN, we note several avoided crossings as the mode frequencies evolve at early times, while the CBMP tracks the modes across the avoided crossings. We verify that the strongest emission of GWs by the PNS corresponds to the -mode in the GCN, indicating that the mode trapping region alternates between the core and the envelope at each avoided crossing. At later times, approximately 0.4 seconds after the bounce, the three classification methods present a similar description of the mode spectrum. We use our results to test universal relations for the PNS modes according to their classification and find that the behaviour of the universal relations for – and -modes is remarkably simple in the CBMP.