We often hear of legendary swords in the medieval European epics – Excalibur, Joyeuse, or Durendal, to name a few. What of other weapons, such as bows and polearms? How did the sword become the high-profile hero's weapon?

There are definitely "legendary," named spears though? I don't know why I wrote that as a question. Odin's got Gungnir, Cu Chulainn has Gae Bolga, in Culhwch and Olwen Arthur has a spear named Rhongomyniad. In the familiar Japanese creation story found in the 古事記 the world was created by the spear 天沼矛. One of Nezha's weapons is the 火尖槍. Guan Yu wields the 青龍偃月刀 in the Romance. In historical times the Three Great Spears of Japan (蜻蛉切, 日本号, and 御手杵), while not strictly legendary, preserve the idea of named, extraordinary spears. I'm not even sure you could say for certain that spears are not "high-profile" heroic weapons. I named some heroes already that have them. Weapons are not named in Homer, but heroes are more defined by their armor, spears, and (especially in the Odyssey of course) bows than by their swords, which are almost an afterthought in Homer--Achilles, after all, wields his spear which is βριθὺ μέγα στιβαρόν ("heavy, huge, thick," to use Lattimore's translation), while his sword when it is drawn is unremarkable. Virgil gives swords somewhat more priority, but when Turnus goes to battle he grabs a spear whose lineage he relates: the spear was taken from the hero Actor, bested in combat, and Turnus exhorts it to kill Aeneas, even though he's literally just girded himself with a sword forged by Vulcan for his father.