  We study the surface quasi-geostrophic equation with an irregular spatial
perturbation $$ \partial_{t }\theta+ u\cdot\nabla\theta =
-\nu(-\Delta)^{\gamma/2}\theta+ \zeta,\qquad
u=\nabla^{\perp}(-\Delta)^{-1}\theta, $$ on $[0,\infty)\times\mathbb{T}^{2}$,
with $\nu\geq 0$, $\gamma\in [0,3/2)$ and $\zeta\in
B^{-2+\kappa}_{\infty,\infty}(\mathbb{T}^{2})$ for some $\kappa>0$. This covers
the case of $\zeta= (-\Delta)^{\alpha/2}\xi$ for $\alpha<1$ and $\xi$ a spatial
white noise on $\mathbb{T}^{2}$. Depending on the relation between $\gamma$ and
$\alpha$, our setting is subcritical, critical or supercritical in the language
of Hairer's regularity structures \cite{Hai14}. Based on purely analytical
tools from convex integration and without the need of any probabilistic
arguments including renormalization, we prove existence of infinitely many
analytically weak solutions in
$L^{p}_{\rm{loc}}(0,\infty;B_{\infty,1}^{-1/2})\cap
C_{b}([0,\infty);B^{-1/2-\delta}_{\infty,1})\cap
C^{1}_{b}([0,\infty);B^{-3/2-\delta}_{\infty,1})$ for all $p\in [1,\infty)$ and
$\delta>0$. We are able to prescribe an initial as well as a terminal condition
at a finite time $T>0$, and to construct steady state, i.e. time independent,
solutions. In all cases, the solutions are non-Gaussian, but we may as well
prescribe Gaussianity at some given times. Moreover, a coming down from
infinity with respect to the perturbation and the initial condition holds.
Finally, we show that the our solutions generate statistically stationary
solutions as limits of ergodic averages, and we obtain existence of infinitely
many non-Gaussian time dependent ergodic stationary solutions. We also extend
our results to a more general class of singular SPDEs.
