Mulligans
General Mulligans
if you get the perfect mulligan (with Jade Claws), Southsea Deckhand is the best entry, but Fire Fly can help in a first turn.
Scorp-o-matic is a good card to keep, because, if you play something like: Southsea Deckhand + Patches, Coin and Fire Fly, and the oponnent play a doomsayer, you have a good remotion.
Jade Claws is a good card to help in removes, and active Charge from Southsea.
Aggro Mulligans
Bloodsail Corsair is a good start versus aggro, because he pull a patches and is a better trader than Soutshea Deckhand, like, he is not dead by any HeroPower, and survivor a minion with 1 Attack.
Scorp-o-matic can kill some cards like, Patches, Bloodsail Corsair, Fire Fly, and the Righteous Protector, and majority of early games.
Maelstrom Portal is a good remove in area.
Midrange Mulligans
Maelstrom Portal is a good way to fight for the board.
Bloodsail Corsair when pulls patches put a good pressure in the first turn.
Scorp-o-Matic can beat Alleycat, Fire Fly, Righteous Protector, Vilefin Inquisitor.
Combo Mulligans
In general combo matches like Highlander Priest, Exodia Mage and most of combo decks need time to buy his cards so:
Scorp-o-matic would be useful against Northshire Cleric for exemple.
Dire Wolf Alpha will put more damage to his face, putting him under pressure, like use Ice Block instead drawing 2 cards in Arcane Intelect.
Control Mulligans
Scorp-o-matic versus control can beat doomsayer, righteous protector, northshire cleric and some early games.
If you start with The Coin, you can keep Flamewreathed Faceless, because its a big pressure on turn 3 (ofc he can get removed by a polymorph, or a shadow word, hex, but in general its a good pressure)
This is my first theory craft for Kobolds and Catacombs, with some tech cards, like: Scorp-o-matic (to kill doomsayer or fire fly, and try to take care of the board), Eater of Secrets will help against Exodia Mage, Leeroy Jenkins to finish the game.