We study spin-wave excitations in a circular ferromagnetic nanodot in different inhomogeneous, topologically nontrivial magnetization states, specifically, vortex and skyrmion states. Gradual change in the strength of the out-of-plane magnetic anisotropy and the Dzyaloshinskii-Moriya exchange interaction leads to continuous phase transitions between different stable magnetic configurations and allows for mapping of dynamic spin modes in and between the vortex, Bloch-type skyrmion, and Néel-type skyrmion states. Our study elucidates the connections between gyrotropic modes, azimuthal spin waves, and breathing modes in these various stable magnetization states and helps us to understand the rich spin excitation spectrum on the skyrmion background.