  We study patch solutions of a family of transport equations given by a
parameter $\alpha$, $0< \alpha <2$, with the cases $\alpha =0$ and $\alpha =1$
corresponding to the Euler and the surface quasi-geostrophic equations
respectively. In this paper, using several new cancellations, we provide the
following new results. First, we prove local well-posedness for $H^{2}$ patches
in the half-space setting for $0<\alpha< 1/3$, allowing self-intersection with
the fixed boundary. Furthermore, we are able to extend the range of $\alpha$
for which finite time singularities have been shown in \cite{KYZ} and
\cite{KRYZ}. Second, we establish that patches remain regular for $0<\alpha<2$
as long as the arc-chord condition and the regularity of order $C^{1+\delta}$
for $\delta>\alpha/2$ are time integrable. This finite-time singularity
criterion holds for lower regularity than the regularity shown in numerical
simulations in \cite{CFMR} and \cite{ScottDritschel} for surface
quasi-geostrophic patches, where the curvature of the contour blows up
numerically. This is the first proof of a finite-time singularity criterion
lower than or equal to the regularity in the numerics. Finally, we also improve
results in \cite{G} and in \cite{CCCGW}, giving local-wellposedness for patches
in $H^{2}$ for $0<\alpha < 1$ and in $H^3$ for $1<\alpha<2$.
