The Supra (rifled) barrel as marketed by the manu is also very useful for shooting European rabbits (oryctolagus cuniculus) while hunting them with a ferret. That was far more common than woodcock hunting.
WC-
Looking at the Verney-Carron catalog from 1999, they list two models with "tube inferieur raye dispersant": Grand Becassier, Plume Becassier. They do add in their text, after some references to woodcock hunting, that the gun is also useful "for short distance shooting of small game". Might be they called them woodcock guns because that sounds more prestigious than rabbit gun.
