Firstly, it's impossible to know for certain how the traditional round shield was used, but we can make a number of assumptions based on evidence from literature (the sagas), the archaeology of construction and wounds suffered in battle and by looking at later fight books such as MS I.33, Talhoffer's duelling shields etc.
Taking the shield discovered in the Gokstad Ship Burial as a base line, we can determine certain characteristics of the Viking-Age Shield.
First, the shield is large: over 35" (90cm) in diameter. Given the average height in 12thC Norway (just after the traditional end of the Viking Age) is around 5'6" (~165 cm), this shield size can potentially cover nearly all of the body of its user.
Secondly, they were thin, around 8-10mm and tapered to the edge. The edge in the Viking Age may have been protected with leather, but there is no evidence of any protection save for bore holes where something may have been attached. The metal clamps of the Vendel period do not seem to have survived into the Viking Age.
Thirdly, the shields were of planked construction, using light-weight wood such as pine and linden.
Finally, they had a small iron boss enveloping the hand and had a centre-gripped handle. The latter point will be particularly important later.
The Teutonic peoples of Northern Europe had a culture of feuding and duelling to settle disputes that lasted far into the middle ages. In Scandinavia, the Holmgang was perhaps the most codified (and perhaps the most well known to us). These duels form a single-combat supremacy amongst warriors, where winning these duels could cement your reputation as a man in the gaze of the gods. Indeed, characters such as Holmgangu Hrafn made a living of duelling.
The use of duelling in Kormaks Saga and Egils Saga shows us the import of this kind of combat, and the Viking feeling towards it.
In the single combat of a duel the shield would be used differently from the use within close-order battle formation. The biggest factor in looking at its use is the grip.
When the grip of the shield is in the centre, the wielder can turn the shield to cover lines of attack on his outside or inside line easily. Thanks to human biomechanics, the use of both lateral and push/pull movements give the wielder great versatility to displace incoming shots. I have deliberately used the word displace rather than block as I see the function of the thinly constructed Viking shield to mitigate the energy of attacks rather than act as a static wall.
This strength and versatility of the centre gripped shield also has a large drawback - a push to the edge of the shield (by a spear for example) can easily turn it in the hand of its wielder. I hypothesize that this why in battle the Vikings fought in their shieldwall of interlocking shields.
Tests done by the folks at Hurstwic show the fragility of the shield against head on attacks.
In single combat, the starting position for most wards would be to cover the line of attack to the head from the outside with the shield turned slightly inwards to displace shots on the shields face. The mostly unprotected and tapered shield rim would sustain significant damage if blows were blocked in any other way - and with the planked construction, they would likely split under a heavy blow. In chapter 150 of Brennu-Njals Saga Kari deflects blows with the face of the shield. On the other hand, such a weak edge seems to have value; in chapter 30 A sword is caught in the edge of Gunnar's shield which he uses (thanks to the centre grip) to twist the sword and snap it. In both cases, the shield is used defensively.
The weapon hand would be out of the way, either protected behind the shield, or behind the body line. In the sword and buckler work of MS I.33, the shield is used to protect the hand in lines of attack, and for the 'shield-knock' or schiltschlac. The shield-knock is a proactive defensive method where the shield is used to bind the opponents sword arm.
In Hand's 'SPADA 2' he demonstrates the use of a shield-knock in the context of a Viking Large Round Shield. Where the opponent attacks in the most obvious way - a swing from his right side to the head of the opponent (the vom-tag or 'from the roof') - the shield edge is thrust forward to meet the weapon-arm shoulder of the opponent, thus binding his arm on the outside of the shield, and prevent the weapon moving over the boss. This maneuver instantly opens options to attack the opponents head.
I think that the shield was used proactively in single combat to displace shots and to bind the opponents weapon-arm.
I do not think it would be used offensively as a weapon to cause damage because of its large size - it would be trivial for an experienced combatant to dodge and use the space provided to attack with their weapon. this isn't to say that it couldn't be used as a weapon, but beyond reasonable doubt, I expect it would be a secondary concern.
In close-order formation, the use of the shield as a weapon would be problematic to the whole formation. In the poem The Battle of Maldon (990AD), the warriors fight in a 'shield wall' which suggests a close order formation of interlocking shield to counter the ease of turning a shield aside with a spear in a 1-on-1 combat. (eg. hold a centre-gripped large shield and have someone push the edge with their pinky - you will not hold it straight!)
The 'Viking Shield' from Archaeology
Hand, S. eds "SPADA 2: Anthology of Swordsmanship" (2010)
Hanson, C. "Population-Specific Stature Reconstruction for Medieval Trondheim, Norway." International Journal of Osteoarchaeology 2 (1992), pp. 289-95.
Hólmgang and Einvigi: Scandinavian Forms of the Duel
Guards of the I.33 or Tower Fightbook - thearma.org/Manuals/I33-guards.html
The story of Burnt Njal (Chapter 150) - sagadb.org/brennu-njals_saga.en#141
Shield Tests - hurstwic.org/history/articles/manufacturing/text/viking_shields.htm