Scabbards would be constructed primarily of thin lathes of wood that would be wrapped in a covering, typically leather. Linen could also be in addition to the leather or a more expensive fabric might be used as a covering. Apart from the covering, they might have decorations in the form of end mounts or metal fittings such as chapes. They might also be lined with a material such as linen or sheepskin.
They were attached to a baldric or belt in order to place them in both a convenient position and angle. Some 14th century swords have additional guards that appear to have been attached to the crossguard of a sword that would help keep the weather out of the scabbard's mouth.
Records from the medieval period and later seem to indicate that when one purchased a sword a scabbard would be included as part of the price unless something very elaborate was required.
In Japan, scabbards were called saya and would be covered with a lacquer and would be used on many weapons such as spears and glaives. Saya for the katana and smaller scabbards would be tucked in the belt, whilst larger swords had mounts attached to a belt or were carried in the hand.