user20563: Suppose all the pieces are on the board. Does there exist a position such that 
A) one of the players can't make any move (a stalemate)?
B) neither player can make any move (a double stalemate)?

user7297: To answer the second part of your question:

Suppose all figures are on the board. Does there exist a transposition of figures such that both of the opponents can't do any move (a stalemate)?

No, this is not possible. Pieces are simply too mobile for this, so for a stalemate you need to hem them in (like the white queen in @RosieF's answer) or pin them on the king. You cannot use pins for a double stalemate, since that would imply the pinning piece can move and it's not stalemate for the other player. (That piece could be pinned itself, but there's no way to make a circular pin.)

user15543: @Rosie F has answered the first part and @im_so_meta_even_this_acronym's answer proves that it is indeed possible with an illegal position. However, I want to focus specifically on legal positions.
The known maximum number of pieces that can be legally stalemated is 30 pieces.  I have two examples.
I found this one in the Die Schwalbe Chess Problem Database..
[Title "Gustavus Charles Reichhelm, Brentano's Monthly 01/1882, 30 Stalemated Pieces"]
[FEN ""]
[startply "50"]

1. Nf3 Nc6 2. Nc3 Nf6 3. Nb5 Ng4 4. h3 a6 5. Na7 Nh2 6. Nxh2 Nxa7 7. g4 b5 8. Bg2 Bb7 9. e4 d5 10. Ke2 Kd7 11. Qg1 Qb8 12. b4 g5 13. Bb2 Bg7 14. Rf1 Rc8 15. Bd4 Be5 16. f3 c6 17. Bf2 Bc7 18. Be1 Bd8 19. Kf2 Kc7 20. a4 h5 21. a5 h4 22. c4 f5 23. c5 f4 24. e5 d4 25. e6 d3

@Laska utilized a promoted piece elsewhere on this site.
[FEN "brnbKRRN/qnk1pBN1/rb1pPpPp/p1pP1P1P/PpP5/1P6/8/8 w - - 0 1"]


user19429: Disclaimer: This solution is not reachable from the starting position, and is not reachable in a game of Chess960 (thanks Rewan!).
 [FEN "3bBNRN/2pPpPKQ/2P1P1PR/7P/p7/rp1p1p2/qkpPpP2/nrnbB3 w - - 0 1"]

Why does the solution here not work?
This is clearly not reachable from the starting position (because of the bishops stuck on the first rank), but the question does not state the position must be legal.

user10380: Here is an example of a 12-move game after which White (to move) is stalemated. All 32 units (pieces and pawns) are still on the board. The original version of this concept game was created by Charles Henry Wheeler, and published in Sunny South in 1887, according to Edward Winter's C.N. 3679. Samuel Loyd is often, and wrongly, given credit.
[Title ""]
[StartFlipped "0"]
[fen "rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR w KQkq - 0 1"]

1.d4 e5 2.Qd2 e4 3.Qf4 f5 4.h3 Bb4+ 5.Nd2 d6 6.Qh2 Be6 7.a4 Qh4 8.Ra3 c5 9.Rg3 f4 10.f3 Bb3 11.d5 Ba5 12.c4 e3 1/2-1/2


20563: <p>Suppose all the pieces are on the board. Does there exist a position such that </p>

<p>A) one of the players can't make any move (a stalemate)?</p>

<p>B) neither player can make any move (a double stalemate)?</p>
 7297: <p>To answer the second part of your question:</p>

<blockquote>
  <p>Suppose all figures are on the board. Does there exist a transposition of figures such that both of the opponents can't do any move (a stalemate)?</p>
</blockquote>

<p>No, this is not possible. Pieces are simply too mobile for this, so for a stalemate you need to hem them in (like the white queen in @RosieF's answer) or pin them on the king. You cannot use pins for a double stalemate, since that would imply the pinning piece can move and it's not stalemate for the other player. (That piece could be pinned itself, but there's no way to make a circular pin.)</p>
 15543: <p>@Rosie F has answered the first part and @im_so_meta_even_this_acronym's answer proves that it is indeed possible with an <strong>illegal</strong> position. However, I want to focus specifically on legal positions.</p>
<p>The <strong>known</strong> maximum number of pieces that can be <strong>legally</strong> stalemated is <strong>30 pieces.</strong>  I have two examples.</p>
<p>I found this one in the <a href="https://pdb.dieschwalbe.de/search.jsp?expression=probid=%27P0002838%27" rel="nofollow noreferrer">Die Schwalbe Chess Problem Database.</a>.</p>
<pre><code>[Title &quot;Gustavus Charles Reichhelm, Brentano's Monthly 01/1882, 30 Stalemated Pieces&quot;]
[FEN &quot;&quot;]
[startply &quot;50&quot;]

1. Nf3 Nc6 2. Nc3 Nf6 3. Nb5 Ng4 4. h3 a6 5. Na7 Nh2 6. Nxh2 Nxa7 7. g4 b5 8. Bg2 Bb7 9. e4 d5 10. Ke2 Kd7 11. Qg1 Qb8 12. b4 g5 13. Bb2 Bg7 14. Rf1 Rc8 15. Bd4 Be5 16. f3 c6 17. Bf2 Bc7 18. Be1 Bd8 19. Kf2 Kc7 20. a4 h5 21. a5 h4 22. c4 f5 23. c5 f4 24. e5 d4 25. e6 d3
</code></pre>
<p>@Laska utilized a promoted piece elsewhere on <a href="https://chess.stackexchange.com/questions/18945/mutual-stalemate">this site.</a></p>
<pre><code>[FEN &quot;brnbKRRN/qnk1pBN1/rb1pPpPp/p1pP1P1P/PpP5/1P6/8/8 w - - 0 1&quot;]
</code></pre>
 19429: <p>Disclaimer: <a href="https://www.chessvariants.com/usualeq.dir/msapansw.html" rel="noreferrer">This solution</a> is not reachable from the starting position, and is not reachable in a game of Chess960 (thanks Rewan!).</p>

<pre><code> [FEN "3bBNRN/2pPpPKQ/2P1P1PR/7P/p7/rp1p1p2/qkpPpP2/nrnbB3 w - - 0 1"]
</code></pre>

<p>Why does the solution here not work?</p>

<p>This is clearly not reachable from the starting position (because of the bishops stuck on the first rank), but the question does not state the position must be legal.</p>
 10380: <p><a href="https://www.chess.com/forum/view/general/stalemate-on-move-12-with-all-the-pieces-on-the-board" rel="noreferrer">Here</a> is an example of a 12-move game after which White (to move) is stalemated. All 32 units (pieces and pawns) are still on the board. The original version of this concept game was created by Charles Henry Wheeler, and published in Sunny South in 1887, according to <a href="http://www.chesshistory.com/winter/winter08.html" rel="noreferrer">Edward Winter's C.N. 3679</a>. Samuel Loyd is often, and wrongly, given credit.</p>

<pre><code>[Title ""]
[StartFlipped "0"]
[fen "rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR w KQkq - 0 1"]

1.d4 e5 2.Qd2 e4 3.Qf4 f5 4.h3 Bb4+ 5.Nd2 d6 6.Qh2 Be6 7.a4 Qh4 8.Ra3 c5 9.Rg3 f4 10.f3 Bb3 11.d5 Ba5 12.c4 e3 1/2-1/2
</code></pre>
