Forty Thieves

If you face a warrior on the ladder, then it's more likely to be a Pirate Warrior than anything else. Since they are quite aggressive, I would advise you to mulligan for defensive tools. Clear their pirates and hope for a good card draw RNG.
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If it's not pirate, then you might face an odd deck. Which is control and thus a favourable MU. If this isn't the case either, then prepare yourself for DMH warrior. Try to dump your hand and play Archbishop Benedictus only after they cast Dead Man's Hand. Keep in mind, that some decks also run Mecha'thun - if your opponent aggressively draws cards, it's a good idea to save the rat for their last turn.

Mech - the only cards, which are good here are: Mass Hysteria,  Bloodmage Thalnos + Spirit Lash, Psychic Scream + Greater Healing Potion, Shadow Visions. Find them, stabilize and win. Clear mechs, since they can be magnetized. Remove the quest.
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Secret - the first thing I want to mention is that Master's Call is super annoying, as the hunter can fetch Barnes and summon a 10/10 Y'Shaarj, Rage Unbound on turn 4. This is why if you have Shadow Word: Death, you should always keep it. It's advisable to do this when you face hunter in general. Secondly - be very careful, when you cast spells. Cat Trick is extremely common, so if you have the possibility, try to cast a spell, which will trigger this secret, and then play a board clear. Save Mass Hysteria for their spellstone. Other than that - stabilize and out-value them.
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(reserved for other decks)

Odd - look for defensive tools, try to save your AoEs (Spirit Lash) when you really need them, i.e. before the paladin plays Quartermaster. If you can afford it, play around Divine Favor.
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Exodia - Congratulations, you are f*cked. Try to build up pressure. If you face this deck too often, add a Dirty Rat.
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Anyfin - don't let the Murloc Warleaders and Old Murk-Eye die. Entomb or Psychic Scream and then Entomb them, if you don't have the this spell beforehand. If somehow the paladin plays smart and they kill their murlocs, play Amara, Warden of Hope before the second Anyfin Can Happen.

Odd or pirate kingsbane. Nothing much to say - just find your defensive tools and try to survive. Hyper aggressive decks are our biggest counter, so it's a good idea to mulligan the quest away, in order to improve your chances of drawing the right cards and thus winning.
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If you stumble upon Big Rogue, try to find Psychic Scream, maybe Lightbomb/Greater Healing Potion and try to maintain your health. You will probably prepare for Odd, but if you somehow know you are facing Big, then keep the quest.

Rare MU. It's mostly jade druid. Either way just try to find reactive stuff so you can deal with their board and out-value them with their own cards.
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Against jade - try save Entomb for Aya Blackpaw. Unless you are low on health and you want to build up Amara, Warden of Hope, I would suggest casting Twilight's Calls after stealing Aya and having her die.

There is a big variety in this class. I would personally advise you to keep the quest in all cases.
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Evenlock - if you somehow know your opponent is playing this deck, keep Shadow Word: Death, Lightbomb and Shadow Visions/Mass Hysteria. The strategy is pretty simple - clear, heal, survive and then out-value them. If you have a lot of health and spare mana, heal your opponent (and don't forget to emote ;P ). This way you play around Hooked Reaver.
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Cubelock - Save Entomb for Mal'Ganis. Same as above - clear, heal, survive and then out-value them. If you somehow suspect a 25-burst combo, then try to time Amara, Warden of Hope correctly.
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Control warlock or Renolock - outvalue them. Play Entomb on Voidlord and Rin, the First Disciple. If they somehow kill her, play Archbishop Benedictus after Azari, the Devourer, this is why you shouldn't play him before Rin enters the board. However some people don't run Shadowflame, so if you have Benedictus and they haven't killed their Rin yet, then play him first. You don't want to risk him being pulled by a Dirty Rat. Oftentimes I purposely don't play any stolen minions from my hand just to minimize the chances of letting my opponent pull Amara or Benedictus.


Some of them run Mecha'thun. RUN THE RAT, if you face this deck too often.

Burn/Tempo - So be really picky, when it comes to the mulligan stage. Take only cheap stuff. Try to play around some secrets - Counterspell, Explosive Runes and Potion of Polymorph/Mirror Entity - by sacrificing cheap spells or high-health minions.
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Combo - there are many variants here. Some use Open the Waygate others just rely on Emperor Thaurissan. In this MU we don't need the quest. Play Dirty Rat (if you run it); if you manage to steal Research Project or Coldlight Oracle, then use them, in order to mill your opponent. In case with the rat - use it 1) after the oppponent played Thauri, 2) or after they played a lot of minions and their quest is almost complete.
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Control (incl. Reno) - "steal, clear, steal, heal, steal, steal, steal and emote".

Your opponent will most likely be Big Priest. The strategy here is pretty simple - steal their annoying minions and narrow down their resurrection pool. This is why I ALWAYS try to find Entomb in my mulligan stage (+ Shadow Visions into a second one or a copy of Psychic Scream). The only minions, which are not worth stealing are Barnes and Obsidian Statue (+ Emperor Thaurissan especially if you have a lot of removals in hand).


The only bad MU I face is the Mind Blast big priest. Against this deck try to Entomb Malygos and Prophet Velen. Don't let them die.
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Razakus - if they play the OTK version, then RIP. Otherwise try to calculate how much damage they can deal in a single turn. 
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Dragon Burn - I have faced it thrice. Same as above - try to calculate how much damage they can deal in a single turn. Don't let them out-tempo you.
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Innerfire Priest - well, this is a bad MU. Try to clear their threats and keep your minions' health as low as possible. If they play both Twilight Acolyte, then you can play threats safely.