The favorite weapon set would be spear and shield, usually with some sort of knife or - if you could afford it - sword in case you needed to get more up-and-personal.

Fighting in the Viking Age relied heavily on shield walls - tight lines of men standing close together and protecting each other with their shields, much like the earlier Greek hoplite phalanx. And just like a hoplite phalanx, the most useful weapon was a spear - it gave you reach, was ideally suited for quick hard stabs, and could (depending on the size of the spearhead) be lighter and quicker than a sword (Viking Age swords tended to be about 2.5lb / 1.1kg; a spear with a moderately sized head and a light ash or hazel shaft could weigh less and be balanced more ideally for a thrust). Most importantly, spears gave each fighter the reach to support the person on either side, and to reach gaps between shields to the right and left instead of being forced to try to get around the shield immediately in front of him. Swords are much less useful in this style of fighting.

The goal of a shieldwall, however, was to push through the other side's formation and get them broken into smaller groups (and hopefully terrify them into running away), at which point swords and long knives (seaxes) might become more useful (though I've seen a fair number of early medieval wounds in the back from spears, so even then they continued to be useful).

And of course, spears can also be thrown. Some were designed specifically to be javelins, but many more appear to have been rather versatile (so you could use it either way). And we have textual accounts of warriors starting off a battle by throwing spears, but keeping one back to use when the fighting became hand-to-hand (the Battle of Maldon is particularly worth a look, if you want to read a great war poem with good descriptions of Viking Age fighting).

Spears were so common that viking age stories like Beowulf refer to especially reliable soldiers as aeswiga ('spear warrior'), and whole people groups as the 'Spear Danes.'

There were other options besides spear and shield - we have people buried with axes that were pretty clearly designed to work better as weapons than tools (some made for use in two hands, toward the end of the Viking Age), and there's a Pictish stone carving showing a warrior with a spear in two hands instead of the more typical one. You don't really see 2-handed swords in this period, however, perhaps because the metallurgy required to make such a long and strong piece of steel was still in its infancy in western Europe, but likely also because the shield remained such an important point of a warrior's social status, enabling him to fight in the shieldwall and, in some early medieval law courts, a required proof of his good social standing.