  We are concerned with the nodal set of solutions to equations of the form
\begin{equation*} -\Delta u = \lambda_+ \left(u^+\right)^{q-1} - \lambda_-
\left(u^-\right)^{q-1} \quad \text{in $B_1$} \end{equation*} where
$\lambda_+,\lambda_- > 0$, $q \in [1,2)$, $B_1=B_1(0)$ is the unit ball in
$\mathbb{R}^N$, $N \ge 2$, and $u^+:= \max\{u,0\}$, $u^-:= \max\{-u,0\}$ are
the positive and the negative part of $u$, respectively. This class includes,
the \emph{unstable two-phase membrane problem} ($q=1$), as well as
\emph{sublinear} equations for $1<q<2$.
  We prove the following main results: (a) the finiteness of the vanishing
order at every point and the complete characterization of the order spectrum;
(b) a weak non-degeneracy property; (c) regularity of the nodal set of any
solution: the nodal set is a locally finite collection of regular codimension
one manifolds up to a residual singular set having Hausdorff dimension at most
$N-2$ (locally finite when $N=2$); (d) a partial stratification theorem.
  Ultimately, the main features of the nodal set are strictly related with
those of the solutions to linear (or superlinear) equations, with two
remarkable differences. First of all, the admissible vanishing orders can not
exceed the critical value $2/(2-q)$. At threshold, we find a multiplicity of
homogeneous solutions, yielding the \emph{non-validity} of any estimate of the
$(N-1)$-dimensional measure of the nodal set of a solution in terms of the
vanishing order.
  The proofs are based on monotonicity formul\ae \ for a $2$-parameter family
of Weiss-type functionals, blow-up arguments, and the classification of
homogenous solutions.
