Hm. Seems to me that

my $n = 10; 
while( $n ){ 
    print $n--;
    next if $n%2;
    ## do other stuff here
} continue{ 
    sleep 1;
}
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Is exactly equivalent to:

my $n = 10; 
while( $n ){ 
    print $n--;
    unless( $n%2 ) {
        ## do other stuff here
    }
    sleep 1;
}
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And far clearer. (Even if you prefer the if( not $n%2 ) form.)

There has to be a good use for it, I'm just not seeing it.

I think the concept is handling complex branching with early return in the main body and exporting operations required every time. How about something more along the lines of:

#!/usr/bin/perl
use strict; 
use warnings;

my @array = qw(1 2 cat dog 1.3 7.2);
for (@array) {
    if (/\./) {
        $_*=2;
        next;
    }
    if (/\d/) {
        $_*=4;
        next;
    }
} continue {
    print;
}
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Of course TIMTOWTDI and I'm trying to come up with a contrived example.

I'm trying to come up with a contrived example.

Yeah! I spent a while trying to contrive something useful too. But not only did I not find anything that couldn't be done without it. I didn't find anything that I could say was even slightly more preferable done with it.

I really had forgotten it existed until it showed up the POD of IO::Socket::Multicast.

---

Examine what is said, not who speaks -- Silence betokens consent -- Love the truth but pardon error.

"Science is about questioning the status quo. Questioning authority".

In the absence of evidence, opinion is indistinguishable from prejudice.

"I'd rather go naked than blow up my ass"

Another difficulty with this structure is that lexicals defined within the  while block are not available in the  continue block:

>perl -wMstrict -le
"my $n = 3;
 while ($n--) {
   my $i = $n * $n;
   }
 continue {
   print qq{n $n  i $i};
   }
"
Global symbol "$i" requires explicit package name at -e line 1.
Execution of -e aborted due to compilation errors.
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I think Perl 6 addresses this through the  KEEP block (?) and related mechanisms.

I, too, cannot remember ever using this structure explicitly, but have use it implicitly occasionally via the  -p command line switch.