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In Birka, Sweden, more than 30 lamellae (individual plates for lamellar armour) were discovered in 1877, 1934, and 1998–2000. They were discovered around the same time as the Gjermundbu mailshirt (900–950), and they could be proof that certain Vikings used this armour, which is made up of a succession of small iron plates threaded together or sewed to a sturdy fabric or leather cats shirt. However, whether the lamellae in question were in the ownership of a Scandinavian resident or a foreign mercenary is a point of contention.
Quilted fabric (a gambeson) has been suggested as a feasible choice for lower-status Viking warriors, despite the fact that there is no mention of it in the sagas. In cemeteries, such things do not endure well, and no archaeological finds have been made. Some runestones appear to show armour, however it is unlikely to be chain mail. It's possible that the armour in question was lamellar armour, or that it wasn't armour at all. Several layers of sturdy linen or hemp canvas, as well as winter clothes made of thick woollen cloth, would give adequate protection at a fair cost. Practical experience with maille suggests that some sort of undergarment would have been worn between the maille and the regular tunic to protect the latter from dirt and excessive wear, but descriptions of the effect of axes in the Sagas indicate that such garments were only lightly padded, if at all.
Leather was significantly more expensive back then than it is now, making it unaffordable for the average fighter. Thorir Hund, the kingsbane, is supposed to have worn a reindeer-fur tunic enchanted by "Finns" (Sámi) to protect him from sword strokes in the Legendary Saga of St. Olaf. The tunic is characterized as "magically" strengthened, implying that it is not a normal example of this type of garment. Leather attire, on the other hand, does occasionally crop up in archaeological digs and would have provided some level of fighting protection.
Overall, the justification for non-metal kinds of armour is still debatable. It's possible that the average Viking fought while wearing everyday clothes and relying solely on his shield for protection.