  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 $3$D 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 $f$-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 $f$- and $p$-modes is remarkably simple in the CBMP.
