I'm not sure why, but the last alternate ((.*)) seems to win in all cases when the other alternates use a look behind. However, things are much easier to understand if you look around less:

use strict;
use warnings;

my $regex = qr/(' ([^']*) ' | " ([^"]*) " | (.*))/x;

do_test (qq~No quote~); 
do_test (qq~'Single quote'~);
do_test (qq~"Double quote"~);


sub do_test {
   my ($line) = @_;

   print "\n";

   if ($line =~ $regex) {
           print "\$1 is $1.\n" if defined $1;
           print "\$2 is $2.\n" if defined $2;
           print "\$3 is $3.\n" if defined $3;
           print "\$4 is $4.\n" if defined $4;
   } else {
      print "No match.\n";
   }
}
[download]

Prints:

$1 is No quote.
$4 is No quote.

$1 is 'Single quote'.
$2 is Single quote.

$1 is "Double quote".
$3 is Double quote.
[download]

Note too various other tidy ups in the code, especially avoiding calling subs with & (which doesn't do what you think) and excessive use of \.