user2470: There are a few different possible ways of capturing an opponent's piece. You could:

Pick up both your piece and their piece simultaneously with both hands, then set your piece down where their piece was.
Pick up both your piece and their piece using the same hand, and then set yours down.
Remove their piece from the board first, and then use the same hand to move your piece.
Pick up your piece, use it to shove theirs out of the way, set down your piece, and then remove their piece.

And perhaps there are a few other ways as well.
Which of these ways are the best etiquette? (The "shove theirs out of the way" method seems particularly inappropriate.) Are any of these ways illegal under tournament rules? (Don't the USCF rules say that a move must be made with one hand?)

user25131: Tournament rules
You say that under USCF rules, each move must be played with one hand only. I do not know if that is true. But under the FIDE Laws of Chess, article 4.1, it is true:

Each move must be played with one hand only.

We can go further than that: you must use that hand to press that clock too (article 6.2.3):

A player must press his clock with the same hand with which he made his move.

Etiquette
The only important points about etiquette are:

Follow the rules.
Do not press the clock using the captured piece (but pressing the clock with the captured piece still in your hand is fine).

Everything else should come automatically, as long as you aim to play efficiently. In moves that involve multiple pieces, handling only one piece at a time (except in castling, where you have to do this) is inefficient. Making a mess is inefficient. Players who make a mess risk moving pieces off the correct squares, which they must fix on their own time (article 7.4.1):

If a player displaces one or more pieces, he shall re-establish the correct position in his own time.

Other answers look to advanced players for examples of how to handle pieces. Naturally, those players are efficient, so they are good examples.

user2001: Updated answer on 5 March 2016:

Take the opponent's piece into your hand.
Place your capturing piece on this square with same hand.
Press clock with same hand.
Place captured piece next to board.
Write down your move on your score sheet.


user70: I have seen the following.  I use technique #1 though as long as one doesn't smash the pieces they are virtually the same.

Pick up both your piece and their piece using the same hand, and
then set yours down. (pick up my piece first, placing it on its new
square, and scooping up the opponent's piece in one motion.)
Pick
up your piece, use it to shove theirs out of the way, set down your
piece, and then move their piece.

Use one hand, and don't take your hand from the board until the entire move is complete.  If you touch one of your pieces, you have to move it.  If you touch one of you opponent's pieces, you have to take it.

user800: Remarkably, I wrote a blog on this topic last night. It is available here - http://www.chess.com/blog/SamCopeland/how-to-move-a-chess-piece.
To quote myself...

Both Magnus and Hikaru  execute their captures by first picking up their opponent's piece, then they slide that piece to the back of their hand to be held by the ring and pinky fingers within their palm. While holding the captured piece, they move their own piece to the capturing square using their thumb, index, and middle finger. Some other players make each move independent. They first remove the captured piece from the board, only then do they move their piece to the capturing square. This can be a noble styling, but it costs precious time in blitz and lightning games.

Regarding legality... Yes, a move must be made with one hand. This is to prevent a player moving a piece and pressing the clock before completing the move. Otherwise, all of the methods you describe are legal, but you are right that using the capturing piece to knock their piece over is rude, and likely to knock pieces over.

user2130: Seeing how Super Grandmasters capture pieces can be instructive:
Garry Kasparov
Capturing an adjacent piece:

Capturing a distant piece:

Source: YouTube
Carlsen (white) vs Caruana

Source: YouTube
Aronian (white) vs Morozevich

Source: YouTube
Hikaru Nakamura (white) vs Vladimir Kramnik

Source: YouTube
Summary:
With the exception of Kramnik, captures proceed as follows:

Lift the opponent's piece and palm it.
With the same hand, move your piece.
Stop the clock using the hand holding the captured piece.
Table the piece.

It's worth noting that it's OK to table the piece in your opponent's time; you do not have to table it before you stop the clock.
Kramnik (playing black in the final image) lifts his own piece before transposing it with Nakamura's pawn in one quick snap. Kasparov also does this with the pawn capture in the first gif. This isn't wrong, it's just a different – and perhaps more aggressive – style. It does not matter if you lift your piece or your opponent's piece first, but all high-level players touch both pieces and the clock with the same hand. (Note that Magnus as white tables the piece on his right even though the clock is on his left.)
It's probably safer not to use two hands for captures or castling, as tournament rules vary on the use of two hands, and it's been a source of controversy in the past. e.g. Giri vs Epishin, Supercube Blitz Utrect, 2009 (scroll down to "The Incident").

2470: <p>There are a few different possible ways of capturing an opponent's piece. You could:</p>

<ul>
<li>Pick up both your piece and their piece simultaneously with both hands, then set your piece down where their piece was.</li>
<li>Pick up both your piece and their piece using the <em>same</em> hand, and then set yours down.</li>
<li>Remove their piece from the board first, and then use the same hand to move your piece.</li>
<li>Pick up your piece, use it to shove theirs out of the way, set down your piece, and then remove their piece.</li>
</ul>

<p>And perhaps there are a few other ways as well.</p>

<p>Which of these ways are the best etiquette? (The "shove theirs out of the way" method seems particularly inappropriate.) Are any of these ways illegal under tournament rules? (Don't the USCF rules say that a move must be made with one hand?)</p>
 25131: <h1>Tournament rules</h1>
<p>You say that under USCF rules, each move must be played with one hand only. I do not know if that is true. But under the <a href="https://handbook.fide.com/chapter/E012018" rel="nofollow noreferrer">FIDE Laws of Chess</a>, article 4.1, it is true:</p>
<blockquote>
<p>Each move must be played with one hand only.</p>
</blockquote>
<p>We can go further than that: you must use that hand to press that clock too (article 6.2.3):</p>
<blockquote>
<p>A player must press his clock with the same hand with which he made his move.</p>
</blockquote>
<h1>Etiquette</h1>
<p>The only important points about etiquette are:</p>
<ol>
<li>Follow the rules.</li>
<li>Do not press the clock using the captured piece (but pressing the clock with the captured piece still in your hand is fine).</li>
</ol>
<p>Everything else should come automatically, as long as you aim to play <em>efficiently</em>. In moves that involve multiple pieces, handling only one piece at a time (except in castling, where you have to do this) is inefficient. Making a mess is inefficient. Players who make a mess risk moving pieces off the correct squares, which they must fix on their own time (article 7.4.1):</p>
<blockquote>
<p>If a player displaces one or more pieces, he shall re-establish the correct position in his own time.</p>
</blockquote>
<p>Other answers look to advanced players for examples of how to handle pieces. Naturally, those players are efficient, so they are good examples.</p>
 2001: <p>Updated answer on 5 March 2016:</p>

<ol>
<li>Take the opponent's piece into your hand.</li>
<li>Place your capturing piece on this square with same hand.</li>
<li>Press clock with same hand.</li>
<li>Place captured piece next to board.</li>
<li>Write down your move on your score sheet.</li>
</ol>
 70: <p>I have seen the following.  I use technique #1 though as long as one doesn't smash the pieces they are virtually the same.</p>

<ol>
<li>Pick up both your piece and their piece using the same hand, and
then set yours down. (pick up my piece first, placing it on its new
square, and scooping up the opponent's piece in one motion.)</li>
<li>Pick
up your piece, use it to shove theirs out of the way, set down your
piece, and then move their piece.</li>
</ol>

<p>Use one hand, and don't take your hand from the board until the entire move is complete.  If you touch one of your pieces, you have to move it.  If you touch one of you opponent's pieces, you have to take it.</p>
 800: <p>Remarkably, I wrote a blog on this topic last night. It is available here - <a href="http://www.chess.com/blog/SamCopeland/how-to-move-a-chess-piece">http://www.chess.com/blog/SamCopeland/how-to-move-a-chess-piece</a>.</p>

<p>To quote myself...</p>

<blockquote>
  <p>Both Magnus and Hikaru  execute their captures by first picking up their opponent's piece, then they slide that piece to the back of their hand to be held by the ring and pinky fingers within their palm. While holding the captured piece, they move their own piece to the capturing square using their thumb, index, and middle finger. Some other players make each move independent. They first remove the captured piece from the board, only then do they move their piece to the capturing square. This can be a noble styling, but it costs precious time in blitz and lightning games.</p>
</blockquote>

<p>Regarding legality... Yes, a move must be made with one hand. This is to prevent a player moving a piece and pressing the clock before completing the move. Otherwise, all of the methods you describe are legal, but you are right that using the capturing piece to knock their piece over is rude, and likely to knock pieces over.</p>
 2130: <p>Seeing how Super Grandmasters capture pieces can be instructive:</p>
<h3>Garry Kasparov</h3>
<p>Capturing an adjacent piece:</p>
<p><img src="https://i.stack.imgur.com/yoxWo.gif" alt="Garry Kasparov captures a piece" /></p>
<p>Capturing a distant piece:</p>
<p><img src="https://i.stack.imgur.com/PKMaG.gif" alt="Garry Kasparov captures another piece" /></p>
<p>Source: <a href="https://www.youtube.com/watch?v=GSqcXPY1Pqo" rel="noreferrer">YouTube</a></p>
<h3>Carlsen (white) vs Caruana</h3>
<p><img src="https://i.stack.imgur.com/Yaf3n.gif" alt="Carlsen and Caruana exchange pieces " /></p>
<p>Source: <a href="https://www.youtube.com/watch?v=x1ZNY7x8K7g" rel="noreferrer">YouTube</a></p>
<h3>Aronian (white) vs Morozevich</h3>
<p><img src="https://i.stack.imgur.com/lsq7a.gif" alt="Aronian and Morozevich exchange pieces" /></p>
<p>Source: <a href="https://www.youtube.com/watch?v=v93vJuxLZGo" rel="noreferrer">YouTube</a></p>
<h3>Hikaru Nakamura (white) vs Vladimir Kramnik</h3>
<p><img src="https://i.stack.imgur.com/ocgoq.gif" alt="Nakamura and Kramnik take pieces" /></p>
<p>Source: <a href="https://www.youtube.com/watch?v=yHrg8Lib0WU" rel="noreferrer">YouTube</a></p>
<h3>Summary:</h3>
<p>With the exception of Kramnik, captures proceed as follows:</p>
<ol>
<li>Lift the opponent's piece and palm it.</li>
<li>With the same hand, move your piece.</li>
<li>Stop the clock using the hand holding the captured piece.</li>
<li>Table the piece.</li>
</ol>
<p>It's worth noting that it's OK to table the piece in your opponent's time; you do not have to table it before you stop the clock.</p>
<p>Kramnik (playing black in the final image) lifts his own piece before transposing it with Nakamura's pawn in one quick snap. Kasparov also does this with the pawn capture in the first gif. This isn't wrong, it's just a different – and perhaps more aggressive – style. It does not matter if you lift your piece or your opponent's piece first, but all high-level players touch both pieces and the clock with the same hand. (Note that Magnus as white tables the piece on his right even though the clock is on his left.)</p>
<p>It's probably safer not to use two hands for captures or castling, as tournament rules vary on the use of two hands, and it's been a source of controversy in the past. e.g. <a href="http://en.chessbase.com/post/playche-in-holland-and-both-hands-in-blitz" rel="noreferrer">Giri vs Epishin, Supercube Blitz Utrect, 2009</a> (scroll down to &quot;The Incident&quot;).</p>
