Alternatively, you could avoid the troubles with a regular expression by setting the record separator $/ to the expression you look for. When you read the file into an array, you get one more elements than the occurences of the string you are looking for:

use strict;
use warnings;

my $constant = '$$';
my $count = -1;
{
        local $/ = $constant;
        $count += () = <DATA>;
}
print "$count\n";

__DATA__
$$ ss $$ ss
$$ ss
ss
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(UPDATE) or as a one-liner:

my $count = do { local $/ = $constant; () = <DATA> } - 1;
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