  We study the singular set in the thin obstacle problem for degenerate
parabolic equations with weight $|y|^a$ for $a \in (-1,1)$. Such problem arises
as the local extension of the obstacle problem for the fractional heat operator
$(\partial_t - \Delta_x)^s$ for $s \in (0,1)$. Our main result establishes the
complete structure and regularity of the singular set of the free boundary. To
achieve it, we prove Almgren-Poon, Weiss, and Monneau type monotonicity
formulas which generalize those for the case of the heat equation ($a=0$).
