In this paper, we consider the degenerate semi-linear Schrödinger and Korteweg-de Vries equations in one spatial dimension. We construct special solutions of the two models, namely standing wave solutions of NLS and traveling waves, which turn out to have compact support, compactons. We show that the compactons are unique bell-shaped solutions of the corresponding PDEs and for appropriate variational problems as well. We provide a complete spectral characterization of such waves, for all values of
p
p
. Namely, we show that all waves are spectrally stable for
2
>
p
≤
8
2>p\leq 8
, while a single mode instability occurs for
p
>
8
p>8
. This extends previous work of Germain, Harrop-Griffiths and Marzuola [Quart. Appl. Math. 78 (2020), pp. 1–32] who have previously established orbital stability for some specific waves, in the range
p
>
8
p>8
.