  Complex solitary waves composed of intersecting vortex lines are predicted in
a channeled superfluid. Their shapes in a cylindrical trap include a cross,
spoke wheels, and Greek $\Phi$, and trace the nodal lines of unstable vibration
modes of a planar dark soliton in analogy to Chladni's figures of membrane
vibrations. The stationary solitary waves extend a family of solutions that
include the previously known solitonic vortex and vortex rings. Their
bifurcation points from the dark soliton indicating the onset of new unstable
modes of the snaking instability are predicted from scale separation for
Bose-Einstein condensates (BECs) and superfluid Fermi gases across the BEC--BCS
crossover, and confirmed by full numerical calculations. Chladni solitons could
be observed in ultra-cold gas experiments by seeded decay of dark solitons.
