See perlvar#$.
rxrx and http://perldoc.perl.org/re.html#%27debug%27-mode and other regex tools
The "anchor" misnomer in regexes (string location assertion)
Why \n matches but not $^?
Disabling regexp optimizations?

matches after newline (or beginning of string). $ matches before newline (or end of string)

$ perl -MData::Dump -Mre=debug -le " dd( $_=qq{a\n\nb} ); s{^$}{boop}m
+; dd( $_ ); "
Compiling REx "^$"
Final program:
   1: MBOL (2)
   2: MEOL (3)
   3: END (0)
anchored ""$ at 0 anchored(MBOL) minlen 0
"a\n\nb"

Matching REx "^$" against "a%n%nb"
   0 <> <a%n%nb>             |  1:MBOL(2)
   0 <> <a%n%nb>             |  2:MEOL(3)
                                  failed...
   2 <a%n> <%nb>             |  1:MBOL(2)
   2 <a%n> <%nb>             |  2:MEOL(3)
   2 <a%n> <%nb>             |  3:END(0)
Match successful!
"a\nboop\nb"

Freeing REx: "^$"
[download]

Trying to match newline after end of line won't work, $\n won't work

$ perl -MData::Dump -Mre=debug -le " dd( $_=qq{a\n\nb} ); s{^$\n}{boop
+}m; dd( $_ ); "
"a\n\nb"

Compiling REx "^%nn"
Final program:
   1: MBOL (2)
   2: EXACT <\nn> (4)
   4: END (0)
anchored "%nn" at 0 (checking anchored) anchored(MBOL) minlen 2
Guessing start of match in sv for REx "^%nn" against "a%n%nb"
Did not find anchored substr "%nn"...
Match rejected by optimizer
"a\n\nb"

Freeing REx: "^%nn"
[download]

But matching an OPTIONAl newline works

$ perl -MData::Dump -Mre=debug -le " dd( $_=qq{a\n\nb} ); s{^$\n?}{boo
+p}ms; dd( $_ ); "
"a\n\nb"

Compiling REx "^%nn?"
Final program:
   1: MBOL (2)
   2: EXACT <\n> (4)
   4: CURLY {0,1} (8)
   6:   EXACT <n> (0)
   8: END (0)
anchored "%n" at 0 (checking anchored) anchored(MBOL) minlen 1
Guessing start of match in sv for REx "^%nn?" against "a%n%nb"
Found anchored substr "%n" at offset 1...
Found /^/m, restarting lookup for check-string at offset 2...
Found anchored substr "%n" at offset 2...
Position at offset 2 does not contradict /^/m...
Guessed: match at offset 2
Matching REx "^%nn?" against "%nb"
   2 <a%n> <%nb>             |  1:MBOL(2)
   2 <a%n> <%nb>             |  2:EXACT <\n>(4)
   3 <a%n%n> <b>             |  4:CURLY {0,1}(8)
                                  EXACT <n> can match 0 times out of 1
+...
   3 <a%n%n> <b>             |  8:  END(0)
Match successful!
"a\nboopb"

Freeing REx: "^%nn?"
[download]