The deck I used to get from 5 to legend over 70 games with 70% winrate.
To see the deck in action, add me on EU server: Twerp#2658
Against aggro decks you will generally be mulliganing for Hydrologist, Aldor Peacekeeper, Dirty Rat, Stonehill Defender.
With coin you can keep Truesilver Champion/Consecration.
Also you can keep Equality/Wild Pyromancer if you got the other one in your hand.
Against slower decks you would like to keep Hydrologist, Dirty Rat, Stonehill Defender and Twilight Drake.
With coin and a good hand, you can also keep Elise the Trailblazer and Stampeding Kodo.
Always be careful with Dirty Rat. If you don't have an answer for the worst case scenario, you simply don't play him.
Class specific mulligans
Druids will in most cases be playing aggressive decks. Mulligan for that.
I would not keep Dirty Rat against Druid, because they usually run either Bittertide Hydra or Finja, the Flying Star.
Having Consecration or Wild Pyromancer/Equality for their Living Mana is very important.
Against Hunter you will need Aldor Peacekeeper to deal with an early Scavenging Hyena. If you don't have Aldor Peacekeeper in your starting hand, you should consider keeping Humility.
As I face mostly freeze mages, I will write a mulligan for those.
Keep Hydrologist and try to get Eye for an Eye, which you will not be playing before you get him to 1 hp. That way he has to Alexstrasza himself to survive. Dirty Rat can be a very important card against freeze mage, since you can take away his Alexstrasza or Archmage Antonidas. Usually you will wait out his smaller carddraw minions before playing the Dirty Rat.
Keep Hydrologist, Truesilver Champion, Stonehill Defender and Twilight Drake. With coin and a good hand, I would sometimes keep Spikeridged Steed and even Sunkeeper Tarim.
Against Priest you would like to snipe his Lyra the Sunshard with Dirty Rat and kill it instantly. If he plays silence Priest, don't hesitate to use an Equality on his Humongous Razorleaf, before he gets to buff it.
In this matchup you have to dig for that Consecration to clear his small minions before he uses his quest. Also you can try to snipe his drawn-back minions with Dirty Rat, and kill them to delay his quest.
For me, the most common Shaman deck on ladder is "flood shaman". Mulligan for all your board clear, and always think of how much damage a Bloodlust would be able to deal to you.
Pretty much the same as Shaman. He will try to fill the board with his small minions. Keep your board clear and don't hesitate to use Wild Pyromancer/Equality combo on turn 4, if he has a decent board.
This is a tricky one, since there's a huge difference between Pirate Warrior and Quest Warrior. Before you mulligan, try to take a look at what cards he is holding over. If he doesn't touch his first card at all, it's probably Quest Warrior. Otherwise mulligan for Pirate Warrior.
Playing the Dirty Rat against Pirate Warrior is a hard decision. If he kept all his cards in mulligan, you should go for it. Otherwise you need to be able to clear his board in the next turn, if something like a Frothing Berserker comes out.
In case of Quest Warrior, you need to play the value game. Try to get as much value out of each card as possible, and sometimes you will have to hold back some cards like Sunkeeper Tarim and Ragnaros, Lightlord and wait for the perfect moment to play them.
I hope you like my Healadin guide and will have just as much fun with it, as I have. Good luck on ladder guys!