Here's the problem: With or without the /m modifier, the regex  /\n$\n/ does not match against the  "\n\n" string!

>perl -wMstrict -le
"my $s = qq{\n\n};
 print $s =~ /(\n$\n)/ ? qq{:$1:} : 'no match';
"
no match

>perl -wMstrict -le
"my $s = qq{\n\n};
 print $s =~ /(\n$\n)/m ? qq{:$1:} : 'no match';
"
no match
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The reason is that the  $\ sequence in the regex is taken as the  $\ 'output record separator' Perl special variable (a newline by default) and interpolated as such in the regex, which thus becomes equivalent to  / \n \n n /x (note the /x modifier).

If the regex is disambiguated as  / \n $ \n /x (again, note the /x modifier), the regex matches both with and without the /m modifier.

>perl -wMstrict -le
"my $s = qq{\n\n};
 print $s =~ /( \n $ \n )/x ? qq{:$1:} : 'no match';
"
:

:

>perl -wMstrict -le
"my $s = qq{\n\n};
 print $s =~ /( \n $ \n )/xm ? qq{:$1:} : 'no match';
"
:

:
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In many of the examples in other replies in this thread, the ambiguity of  $\ in a regex that arises from interpolation is not taken into account and causes (or can cause) confoosion.

Update: Consider the following misleading output from the OP:

string=<a\n>
   no modifier: regex=/^a$\n/
     match => $ matches only boundary, \n matches newline
   [ ... ]
   m modifier (multi line mode): regex=/^a$\n/m
     match => $ matches only boundary, \n matches newline
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In fact, neither regex matches:

>perl -wMstrict -le
"my $s = qq{a\n};
 print $s =~ /^a$\n/  ? '   ' : 'NO ', 'match';
 print $s =~ /^a$\n/m ? '   ' : 'NO ', 'match';
"
NO match
NO match
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The reason for the confusion is that the regex is first defined as  '^a$\n' (i.e., within non-interpolating single-quotes) in the test code, then interpolated within the actual  // regex operator, in which case the  $\ sequence is not ultimately interpolated as the output record separator string.

Again, after appropriate disambiguation, everything's fine:

>perl -wMstrict -le
"my $s = qq{a\n};
 print $s =~ /^ a $ \n/x  ? '   ' : 'NO ', 'match';
 print $s =~ /^ a $ \n/xm ? '   ' : 'NO ', 'match';
"
   match
   match
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