Just a little deeper than the CPAN search there is an exact Class:Singleton DBI implementation (scroll a page or two down):

Class:Singleton DBI Example :)

    package MyApp::Database;
    use vars qw( $ERROR );
    use base qw( Class::Singleton );
    use DBI;

    $ERROR = '';

    # this only gets called the first time instance() is called
    sub _new_instance {
        my $class = shift;
        my $self  = bless { }, $class;
        my $db    = shift || "myappdb";    
        my $host  = shift || "localhost";

        unless (defined ($self->{ DB } 
                         = DBI->connect("DBI:mSQL:$db:$host"))) {
            $ERROR = "Cannot connect to database: $DBI::errstr\n";
            # return failure;
            return undef;
        }

        # any other initialisation...
        
        # return sucess
        $self;
    }
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The above example might be used as follows:

    use MyApp::Database;

    # first use - database gets initialised
    my $database = MyApp::Database->instance();
    die $MyApp::Database::ERROR unless defined $database;
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