  We consider the reconstruction of the support of an unknown perturbation to a
known conductivity coefficient in Calder\'on's problem. In a previous result by
the authors on monotonicity-based reconstruction, the perturbed coefficient is
allowed to simultaneously take the values $0$ and $\infty$ in some parts of the
domain and values bounded away from $0$ and $\infty$ elsewhere. We generalise
this result by allowing the unknown coefficient to be the restriction of an
$A_2$-Muckenhoupt weight in parts of the domain, thereby including singular and
degenerate behaviour in the governing equation. In particular, the coefficient
may tend to $0$ and $\infty$ in a controlled manner, which goes beyond the
standard setting of Calder\'on's problem. Our main result constructively
characterises the outer shape of the support of such a general perturbation,
based on a local Neumann-to-Dirichlet map defined on an open subset of the
domain boundary.
