This guide pertains to both Heroic and Normal mode of Mine Shaft as they are essentially the same thing but Heroic is a bit more aggressive.
The first thing you need to realise is that this encounter is RNG related. If your first few cards are poor then you will have a bad time and if the Mine Shaft rolls too many 6/6 cards then it will also be much harder and in some cases impossible to win. If you are overwhelmed in the first 2/3 turns then concede and go again until RNG favours you. You can play 2 cards per turn (or 1 card 1 hero power) so your starting hand and draws are very important.
Always play Owls/Taunts
Having board control is vital, you should be playing every owl/taunt almost immediately unless you desperately need to clear the existing board. If your starting hand does not contain these cards, it will be much harder to win.
Restart until you get a strong hand
The best cards are the 3/6 owl, the 3/6 taunt and the deal 3 damage to all opponents cards. With these, it is possible to trade efficiently each turn and use your hero power to clear off the rest. On heroic mode, having a copy of Consult Brann (draw 3 cards) on your first turn gives you many more options going into the later turns.
Use the 10 damage on 6/6 enemies
Save your 10 damage cards for the 6/6 creatures. These are the only ones who will trade with your parrots/taunts effectively.
If you are going to hero power, use that first before deciding how to trade. Always try to trade 2 for 1, all your owls should kill 2 enemies (with the help from your hero power).
Save Barrel Forward
Save this card for later turns unless you have no other immediate plays. If you draw this too early then it is essentially a dead card that will cost you the game on Heroic.
How did you beat Mine Shaft?
Comment below and let me know.