user13334: Can Black somehow get a stalemate against the White King-Queen combination or should he try to save his pawn as long as he can?
8/8/5K2/4Q3/4p3/3k4/8/8 b - - 0 1


user7473: Black's only chance to salvage a draw is to of course, keep his pawn with the hope of queening it. However, this is not possible here because after, say, 1...e3 2.Qd5+ Kc2 3. Qe4 Kd2 4.Qd4+, black must block his own pawn with 4...Ke2 and white can play 5.Kf5, bringing his king closer to the pawn. After 5...Kf2 for example, white goes 6.Qf4+ and black needs to go back with 6...Ke2, and white brings his king closer again with 7.Ke4 and plays 8.Qxe3 the next move.
[fen "8/8/5K2/4Q3/4p3/3k4/8/8 b - - 0 1"]

1...e3 2.Qd5+ Kc2 3. Qe4 Kd2 4.Qd4+ Ke2 5.Kf5 Kf2 6.Qf4+ Ke2 7.Ke4 null 8.Qxe3

This idea is similar with the pawn on the 2nd or 7th rank (i.e. 1 square away from queening), except for rook or bishop pawns, which can be drawn under certain conditions (the side with the queen needs to have his king in what most endgame textbooks call the "winning zone"). Wikipedia actually has an article on this. 

user7657: Without going into a lot of detailed analysis, the simplest way to win as White is to get the Q in front of the pawn (1. Qa1 followed by 2. Qe1 will do it) and then bring your K up to win it with the Q's help.  After that it's a simple win with K and Q vs K.  Forget about any stalemate ideas.  They're wishful thinking in this position.

user825: No way. I would resign if I were Black. This is a hopeless position - game over. Anybody other than a total beginner would win for White.
However if:

Black's pawn is a rook or bishop pawn
Black's pawn is close to promotion
Black's king is close to the pawn
White's king is far away from the pawn

It'd be possible to draw by stalemate.

https://chess24.com/en/read/news/7-stalemates-every-chess-player-needs-to-know

Note: colour reversed in the examples.
King + rook pawn vs. queen
7K/7P/6q1/8/8/1k6/8/8 w - - 0 1

King + bishop pawn vs. queen (variation)
7K/5P2/6q1/8/8/3k4/8/8 w - - 0 1


13334: <p>Can Black somehow get a stalemate against the White King-Queen combination or should he try to save his pawn as long as he can?</p>

<pre><code>8/8/5K2/4Q3/4p3/3k4/8/8 b - - 0 1
</code></pre>
 7473: <p>Black's only chance to salvage a draw is to of course, keep his pawn with the hope of queening it. However, this is not possible here because after, say, 1...e3 2.Qd5+ Kc2 3. Qe4 Kd2 4.Qd4+, black must block his own pawn with 4...Ke2 and white can play 5.Kf5, bringing his king closer to the pawn. After 5...Kf2 for example, white goes 6.Qf4+ and black needs to go back with 6...Ke2, and white brings his king closer again with 7.Ke4 and plays 8.Qxe3 the next move.</p>

<pre><code>[fen "8/8/5K2/4Q3/4p3/3k4/8/8 b - - 0 1"]

1...e3 2.Qd5+ Kc2 3. Qe4 Kd2 4.Qd4+ Ke2 5.Kf5 Kf2 6.Qf4+ Ke2 7.Ke4 null 8.Qxe3
</code></pre>

<p>This idea is similar with the pawn on the 2nd or 7th rank (i.e. 1 square away from queening), except for rook or bishop pawns, which can be drawn under certain conditions (the side with the queen needs to have his king in what most endgame textbooks call the "winning zone"). <a href="https://en.wikipedia.org/wiki/Queen_versus_pawn_endgame#Queen_versus_a_pawn_on_the_seventh_rank" rel="nofollow noreferrer">Wikipedia</a> actually has an article on this. </p>
 7657: <p>Without going into a lot of detailed analysis, the simplest way to win as White is to get the Q in front of the pawn (1. Qa1 followed by 2. Qe1 will do it) and then bring your K up to win it with the Q's help.  After that it's a simple win with K and Q vs K.  Forget about any stalemate ideas.  They're wishful thinking in this position.</p>
 825: <p>No way. I would resign if I were Black. This is a hopeless position - game over. Anybody other than a total beginner would win for White.</p>

<p>However if:</p>

<ul>
<li>Black's pawn is a rook or bishop pawn</li>
<li>Black's pawn is close to promotion</li>
<li>Black's king is close to the pawn</li>
<li>White's king is far away from the pawn</li>
</ul>

<p>It'd be possible to draw by stalemate.</p>

<blockquote>
  <p><a href="https://chess24.com/en/read/news/7-stalemates-every-chess-player-needs-to-know" rel="nofollow noreferrer">https://chess24.com/en/read/news/7-stalemates-every-chess-player-needs-to-know</a></p>
</blockquote>

<p><strong>Note:</strong> colour reversed in the examples.</p>

<p>King + rook pawn vs. queen</p>

<pre><code>7K/7P/6q1/8/8/1k6/8/8 w - - 0 1
</code></pre>

<p>King + bishop pawn vs. queen (variation)</p>

<pre><code>7K/5P2/6q1/8/8/3k4/8/8 w - - 0 1
</code></pre>
