  We make rigorous spectral stability analysis for non-resonant
capillary-gravity waves as well as resonant Wilton ripples of sufficiently
small amplitude. Our analysis is based on a periodic Evans function approach,
developed recently by the authors for Stokes waves. On top of our previous
work, we add to the approach new framework ingredients, including a two-stage
Weierstrass preparation manipulation for the Periodic Evans function associated
to the wave and the definition of a stability function as an analytic function
of the wave amplitude parameter. These new ingredients are keys for proving
stability near non-resonant frequencies and defining index functions ruling
both stability and instability near non-zero resonant frequencies. We also
prove that unstable bubble spectra near non-zero resonant frequencies form, at
the leading order, either an ellipse or a circle and provide a justification
for Creedon, Deconinck, and Trichtchenko's formal asymptotic expansion for the
Floquet exponent. For non-resonant capillary-gravity waves for the stability
near the origin of the complex plane, our stability results agree with the
prediction from formal multi-scale expansion. New are our stability results
near non-zero resonant frequencies. As the effects of surface tension vanish,
our result recovers that for gravity waves. Also new are our stability results
for Wilton ripples of small amplitude near the origin as well as near non-zero
resonant frequencies.
