  In this paper we generalize the Local Removable Singularity Theorem in [16]
for minimal laminations to the case of weak $H$-laminations (with $H\in
\mathbb{R}$ constant) in a punctured ball of a Riemannian three-manifold. We
also obtain a curvature estimate for any weak CMC foliation (with possibly
varying constant mean curvature from leaf to leaf) of a compact Riemannian
three-manifold $N$ with boundary solely in terms of a bound of the absolute
sectional curvature of $N$ and of the distance to the boundary of $N$. We then
apply these results to classify weak CMC foliations of $\mathbb{R}^3$ and
$\mathbb{S}^3$ with a closed countable set of singularities.
