  We apply Bridgeland stability conditions machinery to describe the geometry
of some classical moduli spaces associated with canonical genus four curves in
$\mathbb{P}^3$ via an effective control over its wall-crossing. These moduli
spaces with several irreducible components include the moduli space of
Pandharipande-Thomas stable pairs, and the Hilbert scheme, which are not
previously studied. We give a full list of irreducible components of the space
of stable pairs, along with a birational description of each component, and a
partial list for the Hilbert scheme.
