My main is a warrior but my hunter is a total fun char and that is how I treat him. I've done quite a few instances solo and I'd like to share my insights on them.
Here's a armory link:
The World of Warcraft Armory
My normal talent spec for soloing/DPS:
Talent Calculator - World of Warcraft
My pet's talent spec:
Pet Calculator - Wowhead
Now here are a few important things about my setup. I tried a bear for tanking but positioning and swipe threat just doesn't cut it. The perfect tank pet is most likely the gorilla since Thunderstomp produces additional medium threat. I have the complete T0.5 set and depending on the situation I use the head piece fpr 3% more pet health and the chest for 10% pet armor. I've found that even at T7 the T0.5 chest provides superior armor. Same goes for the T0.5 head piece if you buff your pet with stamina. I use the BC world drop pants since those buff your pet a lot and I'm blessed to have a feral staff from Naxx 25 with 714 armor. I also use the Kharazan reputation tanking ring and the Drak'Tharon heroic tanking ring from the last boss. The 2 piece T5 set bonus is also quite essential. I also use 3 pieces of the T1 set on some bosses.
It helps a lot to buff your pet. BC and WotLK pet food provide both the same bonus so Kibler's Bits are the way to go, they provide 30 strength and 30 stamina. Buffing yourself with food also buffs your pet. Also you can use Inscription scrolls to increase yours and your pet's stamina. This is where the T0.5 head becomes powerfull, you can achieve up to 16k-17k hp with your pet when fully self-buffed.
My gorilla has ~22492 armor and 14400+ HP.
Here's a list of what I've done and how:
AQ20
Kurinaxx (Easy, soloed): The healing debuff stacks up to 10 and prevents any pet healing but it doesn't refresh. Once the debuff reaches 10 it runs out and your pet gains full healing.
General Rajaxx (Medium-Hard, soloed): There is one wave where the mini-boss can make you the focus target keep your Feign Death ready for the debuff. Rajaxx himself is also easy the problem is the knockback, it resets threat. It's rather helpfull when your pet has increased run speed and charge to get back to the boss quickly positioning against a wall also helps.
Moam (Easy, soloed): Moam does little damage and you should be able to burn his mana with the new boost to viper sting.
Buru the Gorger (Medium-Hard, not soloed yet): You can prepare an egg with your pet and pull him over, should actually be pretty simple. Haven't tried it too much yet.
Ayamiss the Hunter (Hard-Impossible, not soloed yet): Send your pet in to keep you from getting sacrificed. Since you can't really tank you would need a pet with a ranged attack. No idea if it works haven't tried yet either.
Ossirian the Unscarred (Medium-Hard, not soloed yet): I could imagine the fight to be rather easy, pull your pet to an obelisk and dps when your pet gets in a whirlwind Feign Death.
ZG
Bat-Boss (Medium, soloed): Bomb adds, stand away since the sonic attack silences. Mana burn in pahse 2.
Snake-Boss (Easy, soloed): I recall that you cannot dispell the refresh on the boss so Aimed Shot and Viper Sting both help. In phase 2 just burn the boss down and pull out the pet from the posion clouds.
Spider-Boss (Medium-?, not soloed yet): Not much experience here yet.
Mandokir (Medium, soloed): Kill the Raptor first then then Mandokir. Make sure your pet doesn't have too many sunder armors on. Use Scorpid Sting so that the sunders don't stack too high. Stay on a ledge so that Mandokir doesn't charge, I don't believe this is evade bugging since your pet has always the aggro and Mandokir doesn't charge due to bad game mechanics or rather due to a missing path for a charge, if your pet dies or when you pull threat he will port you down and kill you.
Tiger-Boss (Easy-Medium, soloed): Use Aimed Shot, Viper Sting via macros to reduces the healing effect. Someone already mentioned this earlier, no mana = no heal. I use Aspect of the Viper and do not use Volley, single DPS provides better healing with T5. Use Feign Death or Deterrence when your pet gets stunned. Phase 2 is nothing special.
Panther-Boss (Hard-Impossible, not soloed yet): Volley will keep your pet alive the problem is the debuff that makes you the focus target, the adds will propably kill you in seconds and you cannot stay dead with Feign Death long enough to let it run out. A ferocity pet can maybe tank the boss.
Jin'Do (Easy, not soled yet): The pit should provide enough healing for your pet. Not sure how totems work.
Gahz'ranka (Easy, not soloed yet): Use a scorpid tank pet for the threat reset. I doubt this is hard at all.
Hakkar (No idea, not soloed yet): Going to try soon.
MC
Lucifron (Easy, soloed): Kill the adds with CDs, Bestial Wrath. Make a G15 macro for healing and go watch TV. The curse will kill your DPS, pet DPS wins here.
Magmadar (Easy, soloed): This is where T1 comes in handy. T1 makes your Mend Pet go over 68 yards. Stand out of fear range, app. 40 yards, and let your pet do the work. My brother helped me with the packs, haven't tried them solo yet.
Gehennas (Easy, soloed): Easy with T1, shadow bolts and fire aoe, again, have a 40 yards limit. Use your T1 and go watch TV.
I haven't done more of MC solo and haven't attempted it yet but I could imagine it not to be too hard. Use of T1 and T5 should make it possible. You can equip fire resistance equip for Geddon and with Avoidance your pet should not get too much damage, the T1 should prevent you from getting the bomb. Most special attacks in MC have a max. range of around 40 yards.
BWL
It is possible to solo BWL packs for Elementium Ore. You need T5.
AQ40
It is possible to solo trash in AQ40.
BC
I tried doing Auchenei Crypts on heroic but failed due to too high damage.
EDIT #1:
ZG
Panther-Boss (Easy-Medium, soloed): This boss is rather easy. The trick is to Feign Death when Arlokk is present. You start off the fight with Bestial Wrath and a Mend Pet then instantly Feign Death and stay down until she vanishes. When she's gone get up immediatly and heal your pet and immediatly Feign Death when the CD is ready, even before Arlokk comes back again. Your pet will do 99% of the damage, hit the boss once so you can loot it. Panthers will only attack you when the boss is present and your not in Feign Death mode. Tank the boss at the entrance since at the gong adds will spawn at some point and add on you. When you add one panther all will attack you no matter where they are, this does not count for your pet. When your pet attacks an add it will only fight one add, unless you pull more. I used a ferocity pet with 19.6k armor and 16.7k hp.