user12377: I've heard it said that some Chess960 positions have a significant advantage for White, usually involving fast attacks against undefended pawns. However, since zugzwang positions do exist, it's theoretically possible that Black could have an advantage in some opening setups. Which leads to the question: are there any where he does? If not, are there any that give White less of an advantage than he has in normal chess?

user18809: The answer is probably NO, although there is no way to "prove" that (yet). Zugzwang positions exist, but they appear with many fewer pieces in the board or in some special situations that are definitely not forced from the starting position.
The "statistical evidence" suggested by some answers is not something to take seriously, as samples are small and respond only to how humans play, which is far from perfect (take games by two complete novices and you won't see any advantage for White in classical chess)

user12772: If you accept statistical data taken from games played between computers as evidence, then the answer may be yes.
See this link for the data.
At the time of writing, White scores below 50% in 155 of the 960 positions. A sample of the scores from the RNKBBRNQ starting position is shown below.

Note that the average ratings for both sides are equal, so distortions in expected result due to playing strength should be negligible.

user14079: It happens that someone tried to evaluate all 960 positions with an engine. Several positions got a score of 0.00, but none of them offered any advantage for black. 
Of course, those conclusions are only the evaluations of Stockfish at 40 plies depth. It is a strong hint, but no definite proof, that Black is fine  in the huge majority of 960-chess starting positions, but not better in any of them.
This result matches with the intuition that in a symmetric, non-developed position, with all pawns on the 2d and 7th rank, there is always something useful to do if you are on move: in other words, there cannot be mutual zugzwang.

12377: <p>I've heard it said that some Chess960 positions have a significant advantage for White, usually involving fast attacks against undefended pawns. However, since zugzwang positions do exist, it's theoretically possible that Black could have an advantage in some opening setups. Which leads to the question: are there any where he does? If not, are there any that give White less of an advantage than he has in normal chess?</p>
 18809: <p>The answer is probably <strong>NO</strong>, although there is no way to "prove" that (yet). Zugzwang positions exist, but they appear with many fewer pieces in the board or in <a href="http://www.chessgames.com/perl/chessgame?gid=1102400" rel="nofollow noreferrer">some special situations</a> that are definitely not forced from the starting position.</p>

<p>The "statistical evidence" suggested by some answers is not something to take seriously, as samples are small and respond only to how humans play, which is far from perfect (take games by two complete novices and you won't see any advantage for White in classical chess)</p>
 12772: <p>If you accept statistical data taken from games played between computers as evidence, then the answer may be <strong>yes</strong>.</p>

<p>See <a href="http://www.computerchess.org.uk/ccrl/404FRC/opening_report_by_white_score.html" rel="nofollow noreferrer">this</a> link for the data.</p>

<p>At the time of writing, White scores below 50% in 155 of the 960 positions. A sample of the scores from the RNKBBRNQ starting position is shown below.</p>

<p><a href="https://i.stack.imgur.com/Lm01X.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Lm01X.png" alt="Statistics for RNKBBRNQ starting position"></a></p>

<p>Note that the average ratings for both sides are equal, so distortions in expected result due to playing strength should be negligible.</p>
 14079: <p>It happens that <a href="https://www.chess.com/article/view/whats-the-most-unbalanced-chess960-position" rel="noreferrer">someone tried</a> to evaluate all 960 positions with an engine. Several positions got a score of 0.00, but none of them offered any advantage for black. </p>

<p>Of course, those conclusions are only the evaluations of Stockfish at 40 plies depth. It is a strong hint, but no definite proof, that Black is fine  in the huge majority of 960-chess starting positions, but not better in any of them.</p>

<p>This result matches with the intuition that in a symmetric, non-developed position, with all pawns on the 2d and 7th rank, there is always something useful to do if you are on move: in other words, there cannot be mutual zugzwang.</p>
