To shred cabbage for your smoothie, first, remove any wilted outer leaves and wash the cabbage. Cut the cabbage into quarters and remove the core. Then, using a sharp knife, slice each quarter into thin strips. For a finer texture that blends smoothly, you can use a food processor or a box grater. If using a food processor, pulse the cabbage in short bursts until it reaches the desired consistency. Remember, finer shreds will blend more easily into your smoothie, creating a smoother texture.