  We consider a family of singular surface quasi-geostrophic equations $$
\partial_{t}\theta+u\cdot\nabla\theta=-\nu
(-\Delta)^{\gamma/2}\theta+(-\Delta)^{\alpha/2}\xi,\qquad
u=\nabla^{\perp}(-\Delta)^{-1/2}\theta, $$ on $[0,\infty)\times\mathbb{T}^{2}$,
where $\nu\geq 0$, $\gamma\in [0,3/2)$, $\alpha\in [0,1/4)$ and $\xi$ is a
space-time white noise. For the first time, we establish the existence of
infinitely many non-Gaussian $\bullet$ probabilistically strong solutions for
every initial condition in $C^{\eta}$, $\eta>1/2$ $\bullet$ ergodic stationary
solutions The result presents a single approach applicable in the subcritical,
critical as well as supercritical regime in the sense of Hairer (M. Hairer, A
theory of regularity structures). It also applies in the particular setting
$\alpha=\gamma/2$ which formally possesses a Gaussian invariant measure. In our
proof, we first introduce a modified Da Prato--Debussche trick which, on the
one hand, permits to convert irregularity in time into irregularity in space
and, on the other hand, increases the regularity of the linear solution.
Second, we develop a convex integration iteration for the corresponding
nonlinear equation which yields non-unique non-Gaussian solutions satisfying
powerful global-in-time estimates and generating stationary as well as ergodic
stationary solutions.
