To assign matched substrings you have to use the substitute regex operator
my $s = "Regular expression";
$s =~ s/Regular(.*?)/$1/;
print "s = '$s'\n";
Note that
$s
includes the the whitespace following the matched text, which may not be what you want. But this example should get you going. If you don't want to change the value of $s then the following might be useful
$s = "Regular expression";
($t = $s) =~ s/Regular(.*?)/$1/;
print "s = '$s'\n";
print "t = '$t'\n";
produces
s = 'Regular expression'
t = ' expression'