  We consider a general class of non-diffusive active scalar equations with
constitutive laws obtained via an operator $\mathbf{T}$ that is singular of
order $r_0\in[0,2]$. For $r_0\in(0,1]$ we prove well-posedness in Gevrey spaces
$G^s$ with $s\in[1,\frac{1}{r_0})$, while for $r_0\in[1,2]$ and further
conditions on $\mathbf{T}$ we prove ill-posedness in $G^s$ for suitable $s$. We
then apply the ill/well-posedness results to several specific non-diffusive
active scalar equations including the magnetogeostrophic equation, the
incompressible porous media equation and the singular incompressible porous
media equation.
