Here's how I would re-write this code:

#!/usr/bin/perl -w

use strict;

#  Filter specific strings from an input file into specific files.

{
    #  Open input and output files.

    open ( INPUT, '<', 'html_LogFiles' ) or 
      die "Unable to open input file: $!";


    open ( BSC, '>', 'BSC' ) or die "Unable to open BSC: $!";

    #  Update: Sorry, got the names of these two wrong.
    #  That's why copy and paste is bad, bad, bad.

    open ( SBSCSubsystem, '>', 'SBSCSubsystem' ) or 
      die "Unable to open SBSCSubsystem: $!";
    open ( MCBTSSubsystem, '>', 'MCBTSSubsystem' ) or 
      die "Unable to open MCBTSSubsystem: $!";

    #  Process input lines into output files. Note that an
    #  input line may match more than one pattern and 
    #  therefore may appear in more than one output file.

    while(<INPUT>) {


      if ( /BSC-/ ) {
        chomp;
        my $copy = $_;
        $copy =~ s/BSC-//;
        print BSC "$_ $copy\n";
      }

      if ( /SBSCSubsystem/ ) {

        chomp;
        my $copy = $_;
        $copy =~ s/SBSCSubsystem//;
        print SBSCSubsystem "$_ $copy\n";
      }

      if ( /MCBTSSubsystem/ ) {

        chomp;
        my $copy = $_;
        $copy =~ s/MCBTSSubsystem//;
        print MCBTSSubsystem "$_ $copy\n";
      }
    }

    #  Close input and output files.

    close ( INPUT );

    close ( BSC );
    close ( SBSCSubsystem );
    close ( MCBTSSubsystem );
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One comment at the top of the file, and three more comments after that. Your coding style adds too many comments, which means there's too much to read -- keep it simple.

If I had more time, I'd recommend a hash-based data structure to define the string you're looking for. That way, adding more filters would be a matter of adding an entry to a table, not cutting and pasting a block of code.

Updates 1406, January 5: Sorry, bungled file names of last two output files. Fixed now.