The Arctic is an ocean, so claims there should be resolvable by existing rules concerning 12-mile territorial limits and 200 nautical mile exclusive economic zones. But, the law of the sea allows countries to claim beyond the 200 nautical mile limit if they can prove that their continental shelf extends beyond those zones. This has led to more issues.
There are only five nations with claims in the Arctic: The United States, Russian, Canada, Norway and Denmark (Greenland). There are some claims that are fairly typical, like the sea border area between the Alaska and Canadian territory being in dispute, Hans lsland near Greenland being in dispute between Denmark and Canada, and the question as to whether the Northwest Passage is Canadian territory or international waters. These are all disputes that will probably be solved through diplomacy.
But, confusing the situation is that three nations claim the North Pole. The North Pole is 430 miles (700 kilometers) from the nearest land. The sea depth there is 13,980 feet (4,261 meters).
Canada’s claims go all the way to the North Pole and 200 miles beyond it, based upon where they have claimed that their continental shelf is: Canada Claims North Pole
and: http://www.rcinet.ca/en/2016/05/03/canada-to-submit-its-arctic-continental-shelf-claim-in-2018/
Russia has made similar claims. They are saying the Lomonosov and Mendeleyev ridges are part of its continental shelf and therefore part of its territory. A topographic map of the area is worth looking at:
Denmark also claims the north pole. Apparently the Lomonosov Ridge is also an extension of Greenland.
This map nicely summarizes the confusing and competing claims:
The Russians have gone so far as to dive down to the Lomosonov Ridge and plant a flag there in 2007. It is a flag planed 14,000 feet below sea level.