  We study the patch dynamics on the whole plane and on the half-plane for a
family of active scalars called modified SQG equations. These involve a
parameter $\alpha$ which appears in the power of the kernel in their
Biot-Savart laws and describes the degree of regularity of the equation. The
values $\alpha=0$ and $\alpha=\frac 12$ correspond to the 2D Euler and SQG
equations, respectively. We establish here local-in-time regularity for these
models, for all $\alpha\in(0,\frac 12)$ on the whole plane and for all small
$\alpha>0$ on the half-plane. We use the latter result in [16], where we show
existence of regular initial data on the half-plane which lead to a finite time
singularity.
