Finely shredded cabbage -- the key to good coleslaw -- can be achieved with a sharp knife and a steady hand or with the assistance of a grater, mandolin slicer or blender. Quarter the cabbage beforehand to make pieces of manageable size for any shredding method. Just make sure to remove any wilted leaves from the exterior and thoroughly dry washed cabbage to ensure uniformly crisp and flavorful results.
Remove any wilted or damaged leaves from the exterior of the cabbage and discard. Thoroughly rinse the head under cold, running water and pat dry with paper towels before proceeding. Using a sharp, stainless steel knife, cut the cabbage in half, lengthwise, slicing vertically through the core. Remove the tough core from each half by making two intersecting diagonal cuts just around the core, removing it entirely in a wedge piece. Flip the cabbage halves over so that they lay cut-side down. Cut the halves into quarters, slicing lengthwise. Note: A carbon steel blade will turn the edges of cabbage black.
On a freshly cut cabbage quarter, use a sharp knife and slice downward, running the knife lengthwise along the cabbage into pieces of desired thickness. Try to slice the pieces as thin as you can. Curl your fingers under when you near the end of each slice to protect your fingers from the knife blade.
Position a grater or mandolin over a bowl. Repeatedly slide the cabbage quarter across the edge of the mandolin or grater until you've produced enough shreds. Be careful when shredding using either a utensil, and keep your fingers clear of the sharp surfaces by using a guard or cut-proof gloves.
Place a cabbage quarter in the blender, or, if you wish, cut the cabbage quarter into smaller pieces. Pulse the blender for a few seconds at a time, checking frequently to see if the cabbage is shredded to your preferred thickness. Be careful not to overdo it; otherwise, the cabbage will turn mushy. Remove and repeat with more cabbage until you've shredded enough.