WordPress failed to send email, such as when registering a name or trying to retrieve a password. The problem turned out to be in the wp_mail function in wp-includes/functions.php.
It takes an optional argument $more, which it passes on to mail() as its fifth argument. The php mail() function is documented not to accept that fifth argument when in safe mode. The fix is to call mail() without that $more argument either when in safe mode and/or when $more == ”.
[Update, May 12, 2005] WordPress 1.5.1 has been released and contains the fix to this problem.