  {{In this paper}}, we are concerned with a one-parameter family of peakon
equations with cubic nonlinearity parametrized by a parameter usually denoted
by the letter $b$. This family is called the ``$b$-Novikov'' since it reduces
to the integrable Novikov equation in the case $b=3$. By extending the
corresponding linearized operator defined on functions in $H^1(\mathbb{R})$ to
one defined on weaker functions on $L^2(\mathbb{R})$, we prove spectral and
linear instability on $\LT$ of peakons in the $b$-Novikov equations for any
$b$. We also consider the stability on $\HT$ and show that the peakons are
spectrally or linearly stable only in the case $b=3$.
