  We explore the local well-posedness theory for the 2d inviscid Boussinesq
system when the vorticity is given by a singular patch. We give a significant
improvement of \cite{Hassainia-Hmidi} by replacing their compatibility
assumption on the density with a constraint on its platitude degree on the
singular set. The second main contribution focuses on the same issue for the
partial viscous Boussinesq system. We establish a uniform LWP theory with
respect to the vanishing conductivity. This issue is much more delicate than
the inviscid case and one should carefully deal with various difficulties
related to the diffusion effects which tend to alter some local structures. The
weak a priori estimates are not trivial and refined analysis on
transport-diffusion equation subject to a logarithmic singular potential is
required. Another difficulty stems from some commutators arising in the control
of the co-normal regularity that we counterbalance in part by the maximal
smoothing effects of transport-diffusion equation advected by a velocity field
which scales slightly below the Lipschitz class.
