My current character, Andraste, an Imperial, took the axe with her. She hasn't anywhere near the same level of sentiment as the Nords towards Ysgrammor and his companions, but instead she viewed it as a vital piece of history for the Empire, to whom she is still loyal despite having left the Legion. She couldn't just leave it out there, exposed to the elements, to be eroded away or stolen by bandits. So it is safely on display in Proudspire Manor, until she can figure out a more suitable place for it.