The localized magnon modes of isolated kπ skyrmions on a field-polarized background are analyzed based on the Landau-Lifshitz-Gilbert equation within the terms of an atomistic classical spin model, with system parameters based on the Pd/Fe biatomic layer on Ir(111). For increasing skyrmion order k a higher number of excitation modes are found, including modes with nodes in the radial eigenfunctions. It is shown that at low fields 2π and 3π skyrmions are destroyed via a burst instability connected to a breathing mode, while 1π skyrmions undergo an elliptic instability. At high fields all kπ skyrmions collapse due to the instability of a breathing mode. The effective damping parameters of the spin waves are calculated in the low Gilbert damping limit, and they are found to diverge in the case of the lowest-lying modes at the burst and collapse instabilities but not at the elliptic instability. It is shown that the breathing modes of kπ skyrmions may become overdamped at higher Gilbert damping values.