Putter Hosel & Toe Hang
Also called: putter neck, toe hang
How the shaft connects to the putter head - plumber's neck, slant, double-bend, and center shafts - which sets how much the face rotates during the stroke.
The hosel, or neck, is how the shaft attaches to the putter head, and it determines the putter's toe hang - whether the toe of the putter droops down (toe hang) or the face points to the sky (face-balanced) when you balance the shaft on your finger. This single feature has an outsized effect on how the putter wants to move.
Face-balanced putters, often paired with center-shafted or double-bend hosels, resist face rotation and suit a straight-back-straight-through stroke. Toe-hang putters, paired with plumber's-neck or slant-neck hosels, encourage the face to open and close and suit an arcing stroke. Matching toe hang to your stroke type makes the putter rotate the way your hands already want to.
Hosels also add offset, setting your hands slightly ahead of the face, which helps many golfers return the face square. The amount of offset varies by neck style and is part of why a putter that looks right to one player looks wrong to another.
Putter fitting is often skipped, but matching hosel and toe hang to your stroke - alongside length, loft, and head style - is what makes a putter feel like it swings itself rather than something you have to steer.