  We study the weighted cone transform $I_\kappa$ of distributions with compact
support in a domain $M $ of $\mathbb{R}^3$, over cone surfaces whose vertexes
are located on a smooth surface away from $M$ and opening angles are limited to
an open interval of $(0,\pi/2)$. We show that when the weight function has
compact support and satisfies certain nonvanishing assumptions, the normal
operator $I^*_\kappa I_\kappa$ is an elliptic $\Psi$DO at the accessible
singularities. Then the accessible singularities are stably recoverable from
local data. We prove a microlocal stability estimate for $I_\kappa$. Moreover,
we show the same analysis can be applied to the restricted cone transform.
