A singleton is even easier to do in Perl. You don't even need a reference. Because in Perl you can say

my $class = "Singleton";
$class->method();
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you only need to return a string with the package name in a singleton's constructor.

package Singleton;

my ($various, %instance, @variables);

sub new {
   my $class = shift;

   # object initialization here ... 

   *new = sub { shift }
   return $class;
}

1;
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Here we also avoid the need to check whether the object is initialized by using a glob to replace &new. None of the further calls to the constructor will execute its initial code.

All of your methods can just ignore the $self too, since there's a single instance which stores its data in file lexicals.

Makeshifts last the longest.