  We deal with the local well-posedness theory for the two-dimensional inviscid
Boussinesq system with rough initial data of Yudovich type. The problem is in
some sense critical due to some terms involving Riesz transforms in the
vorticity-density formulation. We give a positive answer for a special
sub-class of Yudovich data including smooth and singular vortex patches. For
the latter case we assume in addition that the initial density is constant
around the singular part of the patch boundary.
