Touch a piece and then claim opponent's previous move was illegal In a last round tie between 2 players fighting for top seed in a competition, here is what happened.

White had 18 seconds on his clock and black had 26 seconds. Black was winning. At move 56, black made a pawn push which exposed his king to a check by a dark square bishop, thus making the pawn push an illegal move. White did not take note so he  raised the bishop and was about to play then he saw that the king was in check, so before he could call 'illegal' black quickly called the 'touch move rule'. Thus implying that white had to move the bishop. According to FIDE rules which call should supersede: Black's 'touch move' or white's 'illegal move'?
In standard play (each player has at least 60 minutes for all the moves) it makes no difference when the illegal moves is spotted the position must be restored to the one before the illegal move even if that was several moves ago.

In rapid or blitz Appendix A4 part 2b applies:


  b. An illegal move is completed once the player has pressed his clock.
  If the arbiter observes this he shall declare the game lost by the
  player, provided the opponent has not made his next move. If the
  arbiter does not intervene, the opponent is entitled to claim a win,
  provided the opponent has not made his next move.


Provided white has not released the bishop on a different square the claim for an illegal move stands and white wins the game.

EDIT: As of 1st January 2018 the rules for rapid and blitz have been brought into line with standard rate. In all 3 cases there is a time penalty for the first illegal move (1 minute for blitz and rapid, 2 minutes for standard) and only the second illegal move loses you the game.