While I know where some of the artifact appearances can be found for other specs, I figured I'd detail the ones I had found so far. Note that alternate colors can be unlocked for each of these by doing 100 Dungeons using the appearance, completing 200 World Quests with the appearance, and killing 1000 enemies in PvP.
The Protection hidden appearance is a Draenei-themed shield and transforms the sword Oathseeker into a crystalline mace. Both the mace and much of the shield are made of purple (by default) crystals, though the Truthguard motif of the spinning central disk and the channels of light energy going right, left, and down are retained.
Getting this appearance requires you to run the Withered Army training scenario for the Nightfallen and it is looted from the main chest at the end (the one you get no matter what.) While I think you may improve your odds if you score higher, it is ultimately a matter of luck (I did several full clears of the scenario with over 30 Withered left at the end - not that they account for that in your score - without getting it to drop.)
Blood (Death Knight):
The Blood hidden appearance transforms your Maw of the Damned into a more one-sided axe, with the edge and the spiked pommel both showing the souls flowing into the axe. There is no skull-face, and the default color is blue, rather than red.
Like the one for Truthguard above, this appearance is acquired by doing Withered Army training (see the above example for more.)
Havoc (Demon Hunter):
There is a Fel Bat that flies above Felsoul Hold in Suramar called Downfall. While you'll be able to reach him if you jump down from other high points (such as Highmountain Peak) the official way to get to him is to find Candrael's charm, a random drop off of the demons in Felsoul Hold. You'll then want to find Candrael, a Blood Elf Demon Hunter who stands on a hill above the western edge of the Hold (you'll want to take a route from the north to reach her.) She will then offer to toss you up in the air at the demon. Downfall creates a pocket of air around himself that allows you to fly, but if you drop below or go too far away from him you'll fall. Soon after engaging him he'll boot you a far distance away and then follow. You'll need to glide in order to get back into his air pocket without falling all the way. As you fight him, he'll throw you away several times again, but he tends to stay in the same place after that first move, so just glide down to him. Oddly, I actually found it easier to fight him as Vengeance, but you'll be able to pick up the item either way. When he dies, he'll plummet to the ground. Glide down to him and loot to get the item that grants the appearance.
The appearance makes the Twinblades of the Deceiver look like they're made of bones and demon fangs, with skull motifs on the handguards. By default there is a fel-green glow around the edges.
Beast Mastery (Hunter):
This is by far the easiest appearance to acquire. It's basically there so that Hunters who prefer bows can get a look for Titanstrike. I wish that they would make gun appearances for Marksmanship and, yes, Survival (ever hear of a bayonet?) but oh well. To get it simply buy it from Hobart Grapplehammer in the Dalaran engineering shop for 8,000 gold.
This transforms Titanstrike into a bow, though thankfully it retains a very mechanical/gnomish look to it.
I don't know exactly what the prerequisites for this are - it seems easier to do now than before 7.1. In Skyhold, talk to the smith (the NPC who lets you reset artifact traits.) He should comment on your sword, wondering if it would be better as an axe. He then asks you to search the Broken Isles for an example of a worthy axe. The search is actually very simple - just go back to Dalaran and you'll find High Overlord Saurfang in Krasus' Landing. Talk to him to try to convince him to help you out (doing this on Alliance will have him comment on how he'd prefer not to empower the Alliance, but that in the face of the Legion, he's willing to do whatever is necessary to protect the world.) He'll then offer to show the axe to the smith if you can best him in combat. Return to Skyhold and jump into the arena off to the side (opposite the area with the smith) and you'll be phased so you can fight Saurfang. It's actually not that tough of a fight. Once you get him to 75% or so, he'll surrender. I don't know if there's an official way to get out of the arena, but if you just jump off a Val'kyr will catch you and take you out of the arena. Go over to the smith and talk to Saurfang, who is waiting there, and you can complete the quest.
This appearance changes Strom'kar to look like a double-bladed axe with a spiked skull-shaped ball on the end of a chain on the pommel.
Other Appearances As I Get Them:
I know of a number of other appearances and even the method of getting them, but right now RNG has not been favoring me for them (frankly I'm pretty happy I have the ones I do.) Many are purely luck-of-the-draw, while other have tricky challenges associated with them while still others require you to reach exalted reputations with various factions. And of course, some have not yet been discovered by the community.