I tried it and it seems to work.

if ($vacation_str =~ m!$HOME/$library::common{username}/!i) { ...
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I have written a little demo to show how to perform regular expression match -

use strict;
use warnings;

my $HOME = '/usr/home';
my %common;
$common{username} = 'roger';

my @str = (
  'cat: cannot open $HOME/$common{username}/filename',
  "Can't open $HOME/$common{username}/filename !",
  "This is some other string" );

foreach my $vacation_str (@str) {
    print "$vacation_str\n";
    # case 1 - match exact string
    if ($vacation_str =~ m!\$HOME/\$common{username}/filename!i)
    # case 2 - match interpolated string
    #if ($vacation_str =~ m!$HOME/$common{username}/filename!i)
    {
        print "match\n";
    } else {
        print "not match\n";
    }
}
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Case 1 output -

cat: cannot open $HOME/$common{username}/filename
match
Can't open /usr/home/roger/filename !
not match
This is some other string
not match
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Case 2 output -

cat: cannot open $HOME/$common{username}/filename
not match
Can't open /usr/home/roger/filename !
match
This is some other string
not match
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