  We introduce a new method for the analysis of singularities in the unstable
problem $$\Delta u = -\chi_{\{u>0\}},$$ which arises in solid combustion as
well as in the composite membrane problem. Our study is confined to points of
"supercharacteristic" growth of the solution, i.e. points at which the solution
grows faster than the characteristic/invariant scaling of the equation would
suggest. At such points the classical theory is doomed to fail, due to
incompatibility of the invariant scaling of the equation and the scaling of the
solution.
  In the case of two dimensions our result shows that in a neighborhood of the
set at which the second derivatives of $u$ are unbounded, the level set
$\{u=0\}$ consists of two $C^1$-curves meeting at right angles. It is important
that our result is not confined to the minimal solution of the equation but
holds for all solutions.
