I'm surprised there aren't more Karazhan posts. I solo Attumen for a shot at the mount most Tuesday evenings, and I soloed through Shade of Aran repeatedly for reputation. Soloing gear is usually what's in my armory profile:
The World of Warcraft Armory - Coulomb @ Uther - Profile
If that link shows some other gear than a weird mix of PVP and PVE, I use 4pc t10 PVE, one PVE trinket, and the rest PVP gear.
DETAIL
Attumen: The 5 pack of trash right before him is almost as hard as the boss. Pull Midnight. Once Attumen appears, pop wolves, fire elemental, Bloodlust, then burn until they form Voltron. Keep using Twin Howl and Stoneclaw Totem to keep Attumen away. In p2, the fight is common sense: Stoneclaw for shield, MW for self heals, etc. A PVP trinket really helps, in case you eat a knockdown while under half health. You can easily OOM from Intangible Presence if careless, and Grounding Totem won't eat it; time Shamanistic Rage accordingly. If you can't get through p1, try using Earth Elemental and save MW procs for Chain Heal centered on it. (Midnight barely tickles)
Moroes: Cake walk. Pop wolves and Bloodlust early, then blow up adds one by one. You should have no trouble. I don't even use an elemental...
Maiden of Virtue: Your fingers will hurt from the button jam, and it's good practice for not using MW early. Otherwise this is so straightforward that I'd call it the easiest solo fight here. Grounding Totem eats her Holy Fire. Hey, beats dailies.
Romulo and Julianne: Decent practice spamming Purge and Wind Shear. I'm not sure what more there is to say. They have very little health, so the hardest part is not actually blowing Julianne up early. Save your wolves for P3, obviously. Keep using Grounding Totem and Wind Shear to deal with Julianne's DoT and heal, respectively. Trinket the stun if below half health.
Wizard of Oz: Simplest kill order is Roar, Tinhead, Dorothee, Strawman. Toto will die on his own due to incidental/cleave damage. Keep a tremor totem down for Roar's fearbombs. Flame shock the scarecrow when it's up. Kill order barely matters, honestly, you're just bursting things down one by one. If there's any trouble, try an Earth Elemental, use MW procs to Chain Heal it, and peel them off one by one. You don't need Grounding Totem or resist totems for this one, it's easier to nuke. (The Crone won't be up for more than 20 seconds, so I doubt you need my tips there)
(I've never had to fight the Big Bad Wolf)
The Curator: Blow up the sparks. Burning the boss will eventually get you overwhelmed, even in gear like mine. It isn't a bad idea to use Grounding Totem to eat a Hateful Bolt. You won't resist all of them, and the surprise burst in P3 can wreck you. Save cooldowns for his 15% Enrage; Wolves plus Shamanistic Rage plus Bloodlust at that point, and then he's yours.
Shade of Aran: This fight is straightforward and easy until he summons elementals. At that point you want an Earth Elemental and spirit wolves, and you need them tanking as many as you can in case you get a bad combination (e.g. elemental during Flame Wreath). The wolves can (barely) kill 2-3 elementals themselves. Burn them when you can. (If you feel the need to lock a school out, just pick one)
I have nothing major to add about ZG or MC, though I've soloed parts of both. The mount bosses are cake as Enh, though maybe that's my gear. Here's one alternate strategy; what I do for High Priest Thekal is identical to what's been discussed already.
Bloodlord Mandokir: It's not that hard to fight him in the usual place. Start with an Earth Elemental to deal with the raptor so you don't eat sunders. Pop wolves and bloodlust. After the first fear, assuming you don't get feared in identical directions, Mandokir will be on you, Ohgan on your elemental. If not, keep peeling him off and keep them apart. You have two minutes to blow him up before you start eating sunders. The occasional MW heal on the elemental helps a lot.
My questions:
1. The only part of ZG that gives me trouble are the trash packs before Hakkar. Channeled mindflay stun = ow. How to deal with that solo? Can those packs be skipped?
2. MC, I can't figure out how to deal with Geddon (even with Water Shield, I run oom and die to Ceiling Boss), or Sulfuron Harbinger. For the latter, how do you deal with his healers? They blow up my elemental and heal too fast for my burst.