Regular and bandana styles can be laid flat, before tucking in the two arms and then folding again into halves or quarters. You could also fold all bibs in half lengthways and then roll them up. Store folded or rolled bibs in a pretty shoe box, or use a wardrobe organizer to keep them neat and in one place.
Smock or full coverage bibs can be folded as you would any other shirt or apron, but take care to make sure that plastic materials are completely dry before storing them away. Mold can develop if plastic bibs are packed away before they’re properly dry – yuck!