user12130: What are some common tactics or ideas that arise when black has played h6 (or white has played h3) in front of his castled king?
An example I have in mind is, in this position, white can win a pawn with Bxh6, gxh6, Qxf6:

A less concrete but still important idea is, with a bishop on the a2-g8 diagonal (pinning black's pawn on f7), white often plans to sink his king's knight into g6 in a couple of moves:


user11629: In my experience, your second  idea, sinking a knight in g6, is not very common. Some more ideas:

piece sacrifice on h6 for two pawns and an open king
piece sacrifice on g5 (not forced but tempting). If white has a pawn on h4 and rook on h1, the opening of the h file can often be very dangerous for black. This is also known as fishing pole trap


user7297: Two basic ideas:

Put a knight on f5; Black will have a hard time chasing it away with g6 without losing material.
Storm with your g-pawn to g5. Especially with the black knight still on f6, Black cannot avoid the opening of some lines.

Both plans would work in some circumstances with the pawn still on h7, but are a lot more effective with the pawn on h6.

12130: <p>What are some common tactics or ideas that arise when black has played h6 (or white has played h3) in front of his castled king?</p>

<p>An example I have in mind is, in this position, white can win a pawn with Bxh6, gxh6, Qxf6:</p>

<p><img src="https://i.stack.imgur.com/pDu6E.jpg" alt="white can win a pawn"></p>

<p>A less concrete but still important idea is, with a bishop on the a2-g8 diagonal (pinning black's pawn on f7), white often plans to sink his king's knight into g6 in a couple of moves:</p>

<p><img src="https://i.stack.imgur.com/bOzFR.jpg" alt="g6 is weak"></p>
 11629: <p>In my experience, your second  idea, sinking a knight in g6, is not very common. Some more ideas:</p>

<ul>
<li><p>piece sacrifice on h6 for two pawns and an open king</p></li>
<li><p>piece sacrifice on g5 (not forced but tempting). If white has a pawn on h4 and rook on h1, the opening of the h file can often be very dangerous for black. This is also known as <em>fishing pole trap</em></p></li>
</ul>
 7297: <p>Two basic ideas:</p>

<ol>
<li>Put a knight on f5; Black will have a hard time chasing it away with g6 without losing material.</li>
<li>Storm with your g-pawn to g5. Especially with the black knight still on f6, Black cannot avoid the opening of some lines.</li>
</ol>

<p>Both plans would work in some circumstances with the pawn still on h7, but are a lot more effective with the pawn on h6.</p>
