Regarding:
        ...it WILL stop matching on the first line.
       You gotta let 'er match the entire string.

Nope. (Did you test your assertion before posting?)

Viz:

     my $s = "line1\r\nline2\r\nline3\r\n";
     print $s;
     $s =~ s/[\n\r]//g;
     print $s;
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Prints:

     line1
     line2
     line3
     line1line2line3
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It's good that you are alert to the pitfalls of matching in multi-line strings. (They can catch you off guard.) But your assertion perpetuates a common confusion that is only confusing if you let it be.

Here's what both perlman:perlre and/or a quick test script will show you:

   /s simply allows . to match end-of-line chars 
       anywhere in the string (if needed)
   /m simply allows ^ and $ to match begin/end
       of lines in mid-string instead of only
       the begin/end of the entire string.
   /g is happy to repeat searching over several lines
  m// (simple search) will find something that
       occurs on a line after the first one
   \n
   \r will both be found (if present) without
       resorting to /s or /m
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