You should also enable warnings (and strictures; see strict), especially if you are a Perl novice. Consider your first regex with warnings enabled:

c:\@Work\Perl\monks>perl -le
"use warnings;
 use strict;
 ;;
 my $s = qq{aaa       : AAA\n}
       . qq{bbb       : BBB\n}
       . qq}ccc       : CCC\n}
       ;
 print qq{[[$s]]};
 ;;
 my $m = 'bbb';
 ;;
 my $t = $s =~
   s/.*^$m *: (.*?)$.*/$1/rsm
   ;
 ;;
 print qq{[[$t]]};
"
[[aaa       : AAA
bbb       : BBB
ccc       : CCC
]]
Use of uninitialized value $. in regexp compilation at -e line 1.
[[BBB
ccc       : CCC
]]
[download]

The Use of uninitialized value $. in regexp compilation... message gives you a clue about what is happening.

If the  $ is unambiguously a regex metacharacter:

c:\@Work\Perl\monks>perl -le
"use warnings;
 use strict;
 ;;
 my $s = qq{aaa       : AAA\n}
       . qq{bbb       : BBB\n}
       . qq}ccc       : CCC\n}
       ;
 print qq{[[$s]]};
 ;;
 my $m = 'bbb';
 ;;
 my $t = $s =~
   s/.*^$m *: (.*?)$(?:.*)/$1/rsm
   ;
 ;;
 print qq{[[$t]]};
"
[[aaa       : AAA
bbb       : BBB
ccc       : CCC
]]
[[BBB]]
[download]

You have your intended output for this regex.