My favorite is a solid core variation called the Archimedean worm, an auger of metal thread spiraling down a tapering rod. It is so called because in silhouette it resembles the great inventor's 200 B.C. waterscrew. I grieve that this variant never seems to be found on pocket knives. Sometimes grooves, single or double, are seen on the outside of the worm or helix. Grooves ease cork penetration and improve grip.
Grooves look good, too.
Double grooves or fluting are especially desired. The biting screw is turned clockwise for right-handed users, and counterclockwise for lefties. I didn't know handedness made a difference. Nobody ever asked me what I needed. Maybe that's why I mangle so many corks.