  We consider the SQG equation with dissipation given by a fractional Laplacian
of order $\alpha<\frac{1}{2}$. We introduce a notion of suitable weak solution,
which exists for every $L^2$ initial datum, and we prove that for such solution
the singular set is contained in a compact set in spacetime of Hausdorff
dimension at most $\frac{1}{2\alpha} \left( \frac{1+\alpha}{\alpha} (1-2\alpha)
+ 2\right)$.
