  We construct solutions in $\mathbb{R}^2$ with finite energy of the surface
quasi-geostrophic equations (SQG) that initially are in $C^k$ ($k\geq 2$) but
that are not in $C^{k}$ for $t>0$. We prove a similar result also for $H^{s}$
in the range $s\in(\frac32,2)$. Moreover, we prove strong ill-posedness in the
critical space $H^{2}$.
