  We study singularity formation for the focusing quadratic wave equation in
the energy supercritical case, i.e., for $d \geq 7$. We find in closed form a
new, non-trivial, radial, self-similar blowup solution $u^*$ which exists for
all $d \geq 7$. For $d=9$, we study the stability of $u^*$ without any symmetry
assumptions on the initial data and show that there is a family of
perturbations which lead to blowup via $u^*$. In similarity coordinates, this
family represents a co-dimension one Lipschitz manifold modulo translation
symmetries. In addition, in $d=7$ and $d=9$, we prove non-radial stability of
the well-known ODE blowup solution. Also, for the first time we establish
persistence of regularity for the wave equation in similarity coordinates.
