That's what hashes are good at. Since keys have to be unique, just plop your parsed version of $mer into a hash as the hash's key. Value is unimportant, though you could use it as a counter.

Example just teaking your code slightly:

use strict;
use warnings;
open (MER, "Extracted.c") or die "Can't open input file. $!\n";
my %cases;
for my $mer(<Mer>){
    my $control;
    while ($control!=15){
        $mer =~ s/ //; 
        $control++;
    }
    $cases{$mer}++;
}
foreach $key ( keys %cases ) {
    print $key, "\n";
}
[download]

I'm not really clear on that the inner loop is for. You only want to remove the first fifteen spaces from $mer? ...ok. I guess that's working out ok. You could alternatively use something like:

substr($mer,0,15) =~ s/ //g;

That would eliminate the while loop and $control counter.