The speed difference is quite small and probably not worth worrying about in most cases. That being said, double quotes are a bit slower mainly because the parser has to look for variables within the string, and replace them with their values when it finds them. (It also has to do a similar replacement for special escape sequences such as "\n" for a newline.)
So, if you do the following, the single-quoted version will be faster (though maybe only by a micro-second or two):
$text = 'This is a test.'; // a bit faster than:
$text = "This is a test.";
But, if variables are involved, I'm not sure which would be faster:
$var = 'test';
// which is faster? (I don't know.):
$text = "This is a $var.";
$text = 'This is a'.$var.'.';
$trext = sprintf('This is a %s.', $var);