  In this paper we are concerned with singular points of solutions to the {\it
unstable} free boundary problem $$ \Delta u = - \chi_{\{u>0\}} \qquad \hbox{in}
B_1. $$ The problem arises in applications such as solid combustion, composite
membranes, climatology and fluid dynamics.
  It is known that solutions to the above problem may exhibit singularities -
that is points at which the second derivatives of the solution are unbounded -
as well as degenerate points. This causes breakdown of by-now classical
techniques. Here we introduce new ideas based on Fourier expansion of the
nonlinearity $\chi_{\{u>0\}} $.
  The method turns out to have enough momentum to accomplish a complete
description of the structure of the singular set in ${\mathbb R}^3$.
  A surprising fact in ${\mathbb R}^3$ is that although
$$\frac{u(r\x)}{\sup_{B_1}|u(r\x)|}$$ can converge at singularities to each of
the harmonic polynomials $$ xy, {x^2+y^2\over 2}-z^2 \textrm{and}
z^2-{x^2+y^2\over 2},$$ it may {\em not} converge to any of the
non-axially-symmetric harmonic polynomials $\alpha((1+ \delta)x^2 +(1-
\delta)y^2 - 2z^2)$ with $\delta\ne 1/2$.
  We also prove the existence of stable singularities in ${\mathbb R}^3$.
