  We study singularities of spacelike, constant (non-zero) mean curvature (CMC)
surfaces in the Lorentz-Minkowski 3-space $L^3$. We show how to solve the
singular Bj\"orling problem for such surfaces, which is stated as follows:
given a real analytic null-curve $f_0(x)$, and a real analytic null vector
field $v(x)$ parallel to the tangent field of $f_0$, find a conformally
parameterized (generalized) CMC $H$ surface in $L^3$ which contains this curve
as a singular set and such that the partial derivatives $f_x$ and $f_y$ are
given by $\frac{\dd f_0}{\dd x}$ and $v$ along the curve. Within the class of
generalized surfaces considered, the solution is unique and we give a formula
for the generalized Weierstrass data for this surface. This gives a framework
for studying the singularities of non-maximal CMC surfaces in $L^3$. We use
this to find the Bj\"orling data -- and holomorphic potentials -- which
characterize cuspidal edge, swallowtail and cross cap singularities.
