  We prove that the focusing cubic wave equation in three spatial dimensions
has a countable family of self-similar solutions which are smooth inside the
past light cone of the singularity. These solutions are labeled by an integer
index $n$ which counts the number of oscillations of the solution. The
linearized operator around the $n$-th solution is shown to have $n+1$ negative
eigenvalues (one of which corresponds to the gauge mode) which implies that all
$n>0$ solutions are unstable. It is also shown that all $n>0$ solutions have a
singularity outside the past light cone which casts doubt on whether these
solutions may participate in the Cauchy evolution, even for non-generic initial
data.
