I need the script to give me as a result the line of S12 as the lowest column3 value

I presume you actually meant you want the later line in the event of ties. (Upd: No, I don't think that's what you meant. I don't understand what you do want, so I can't offer another solution. )

$_ = <>;
my @fields = split;
my $lowest = my $highest = $fields[2];

while (<>) {
   my @fields = split;
   if ($fields[2] <= $lowest) {
      $lowest = $fields[2];
   }
   if ($fields[2] >= $highest) {
      $highest = $fields[2];
   }
}

print("$lowest\n");
print("$highest\n");
[download]

since S12 has value 1 appear more often than S11 in column3 based on the value in column1

How do you get that? As far as I can tell, in your example data, 1 occurs exactly once in column 3 for both S11 and S12, and exactly twice in column 1 for both S11 and S12. So what is the discriminator really? And what did you mean by "based on"? That's exceedingly vague.

My best guess as to your meaning is that column 1 is the frequency of the value in column 3.

Here's one way that might be coded:

#! perl -slw
use strict;

my @data;
push @data, [ split ] while <DATA>;

@data = sort {
    $a->[ 0 ] * $a->[ 2 ]
    <=>
    $b->[ 0 ] * $b->[ 2 ]
} @data;

print "Lowest: ". join "\t", @{ $data[ 0 ] };
print "Highest:", join "\t", @{ $data[-1 ] };

__DATA__
2870      S11      1
574       S11      2
317       S11      3
31        S11      4
1         S11      6
1         S11      7
2925      S12      1
8         S12      5
1         S12      6
1         S12      9
[download]

And the output is:

c:\test>junk37
Lowest: 1       S11     6
Highest:2925    S12     1
[download]

As both groups of data have 1 x 6 values, either (or both) are candidates for the 'lowest', so you'll have to decide which through some further qualification to the sort: eg. you might consider that in the event of a tie S11 is 'lower' than S12. Or, you might decide to display both (all) equally low values.

Of course, if the set of values is large, especially if it is larger than memory, sorting is an inefficient way of find the largest and smallest. The main point is that (assuming I understood you correctly) that you need to multiply column 1 by column 3 when doing your comparisons.

Feel free to ask questions about this code. It is easier to answer specific questions than guess what you might not know.

Another way to do this by reading the file a line at a time:

$ echo "Column1 Column2 Column3

2870      S11      1
574       S11      2
317       S11      3
31        S11      4
1         S11      6
1         S11      7
2925      S12      1
8         S12      5
1         S12      6
1         S12      9
" | perl -e'
use warnings;
use strict;

my ( @lowest, @highest );

while ( <> ) {
    next if /^\D/;
    my @fields = split;
    unless ( @lowest ) {
        @lowest = @highest = @fields;
        next;
        }
    if ( $lowest[ -1 ] >= $fields[ -1 ] && $lowest[ 0 ] < $fields[ 0 ]
+ ) {
        @lowest = @fields;
        }
    if ( $highest[ -1 ] < $fields[ -1 ] ) {
        @highest = @fields;
        }
    }
print join( "\t", "Lowest:", @lowest ), "\n";
print join( "\t", "Highest:", @highest ), "\n";
'
Lowest: 2925    S12     1
Highest:        1       S12     9
[download]

Thanks for all of your replies. Yes, column 1 shows the frequency of the value in column 1. Could you please show me how to load my txt.file into your script, so that I can actually try it. I tried to do it the way I did it with my last script, but it somehow doesn't work. And as mentioned I am such a beginner in Perl that I don't know any other ways of how to upload txt files yet... Thanks so much to all of you for trying to help me!

If you see a loop like

while (<>){
 ...
}
[download]

then it means that the script is reading from STDIN (standard input). You now need to find out how to compose a command line that passes your file on STDIN to the perl script, which is a nice little exercise... :)
One of the solutions posted here used a DATA block at the end of the script, i.e. there is a signal to tell the perl interpreter that the rest of the file is not Perl code but data and the while loop reads from the data block line by line instead of STDIN.