Please note almost all Arctic fox color morphs are man made. The only naturally occurring ones are albino, polar, and dark Arctic foxes.

Polar Arctic foxes. These have a seasonal coat and in the summer (along with when they are kits) they have a black/grey/brown coat, when in the winter they have pure white fur

Blue Arctic fox.

Dark Arctic fox

Blue star Arctic fox.

The Haugen platinum arctic fox

The tundra Arctic fox.

The lapponia Arctic fox. Also known as the pearl lapponia Arctic fox and the batonia pearl Arctic fox.

The sognilibles shadow arctic fox is a rare colour mutation of the arctic fox which is all white. The kits are born white and stay this colour year long, whether it's winter or summer. Their eyes, based on documentation and pictures available of this mutation, seem to always be blue and their nose, bright pink. These foxes are said to have smaller eyes as well as longer, more narrow muzzles than any other mutation of the arctic fox. (Quote from a previously mentioned website)

The lapponia sognilibles Arctic fox.

The sognilibles Arctic fox

The Swedish sapphire Arctic fox

Finnish sapphire Arctic fox

Polar sapphire Arctic fox. These are rare and extremely hard to produce.

Oppdal Arctic fox. It's also known as the Norwegian sapphire Arctic fox

Shadow arctic fox (white, pale blue, dark blue) (Ss). These Arctic foxes stay a similar/the same color year round, no matter what season it is.

Pearl Arctic fox. These have three different categories pale pearl, polar pearl, and dark pearl.

Nordic blue Arctic fox. These look a lot like normal blue Arctic foxes except these are very pale in comparison and the markings are rearranged differently.

Greenland blue Arctic foxes. They are also known as Arctic blue Arctic foxes.

Piebald Arctic foxes. They are also known as spotted Arctic fox due to their markings. Sadly I doubt there are more than about 20 something therians with this theriotype as it is one of the most rare color morph and the spotted gene was never really passed down. Livingwithfoxes.com does a better explanation of this

"Albino arctic fox
(cc)

The albino arctic fox is a naturally occurring mutation of the arctic fox. They are all white all year long, regardless of the season. Since they completely lack pigmentation, their nose is pink and their eyes appear either blue or pink. They are born white.
Shadow sapphire arctic fox (Ssff)

Shadow sapphire arctic foxes are a mutation of the arctic fox which resembles, in appearance, shadow foxes. The main difference is that, rather than having grey tints (for the individuals that are dark or medium coloured), they have blue tints. Their nose is grey and their eyes, generally yellow or brown, although they can be blue, partially coloured or bi-eyed as well.
Lapponia shadow (LlSs)

Lapponia shadow arctic foxes are a mutation of the arctic fox which resembles shadow arctic foxes in appearance. When they are in their winter coat, they are undistinguishable from pale shadow arctic foxes or white shadow arctic foxes. However, during summer, they can be distinguished from lighter shadow foxes by the slightly more blueish tints of the few pale markings on their body. Their eyes are brown, yellow, bi-eyed or blue and their nose, black. They can be bred to shadow foxes without risking embryonic mortality as they are not an allele of the shadow gene, unlike jotun foxes and haugen platinum foxes.
Sognilibles shadow (SsTt)

Sognilibles shadow arctic foxes are a mutation of the arctic fox which resemble shadow arctic foxes in appearance. The main difference is that the patterns on the darker individuals are irregular, just like in sognilibles foxes. Their eyes can be blue, bi-eyed, partially coloured or brown. Their noses are black." (Quotes from the previously mentioned website)