First some facts. I once looked this stuff up because when I was a kid, I was try8ing to figure out which nationality Santa Claus would be. It happens to be the case that the northernmost point on land in Greenland is 440 miles from the North Pole, the northernmost point on land in Canada is 472 miles from the North Pole, and the northernmost point on land in Russia is 493 miles from the North pole.
Canada and Russia are both independently sovereign, which I think gives their claims to the pole more credibility than Denmark's. However, Russia's claim over the territory is weaker, IMO, since the pole is actually on the North American continental shelf, not part of Eurasia at all. Also, for what it's worth, the northernmost populated settlement happens to be located in Canada.
However, national borders do not extend any further than about 14 miles into the ocean (basically, approximately the distance to the horizon as seen from a tall ship's crows nest) so in the end, I think none of the countries have any true claim over the territory in terms of their national jurisdiction.