  In the present work we revisit the $b$-family model of peakon equations,
containing as special cases the $b=2$ (Camassa-Holm) and $b=3$
(Degasperis-Procesi) integrable examples. We establish information about the
point spectrum of the peakon solutions and notably find that for suitably
smooth perturbations there exists point spectrum in the right half plane
rendering the peakons unstable for $b<1$. We explore numerically these ideas in
the realm of fixed-point iterations, spectral stability analysis and
time-stepping of the model for the different parameter regimes. In particular,
we identify exact, stationary (spectrally stable) lefton solutions for $b<-1$,
and for $-1<b<1$, we dynamically identify ramp-cliff solutions as dominant
states in this regime. We complement our analysis by examining the breakup of
smooth initial data into stable peakons for $b>1$. While many of the above
dynamical features had been explored in earlier studies, in the present work,
we supplement them, wherever possible, with spectral stability computations.
