  This work is devoted to the geometric-theoretic analysis of the nodal set of
solutions to degenerate or singular equations involving a class of operators
including $$ L_a = \mbox{div}(\abs{y}^a \nabla), $$ with $a\in(-1,1)$ and their
perturbations.
  As they belong to the Muckenhoupt class $A_2$, these operators appear in the
seminal works of Fabes, Kenig, Jerison and Serapioni \cite{fkj,fjk2,fks} and
have recently attracted a lot of attention in the last decade due to their link
to the localization of the fractional Laplacian via the extension in one more
dimension \cite{CS2007}. Our goal in the present paper is to develop a complete
theory of the stratification properties for the nodal set of solutions of such
equations in the spirit of the seminal works of Hardt, Simon, Han and Lin
\cite{MR1010169,MR1305956,MR1090434}.
