ikegami's solution will work nicely if you only have one such pattern per line. If you have more than one such pattern, you might want to try using the g modifier with your regex. This lets you repeated search for a pattern in a line, like this:

use strict;
use warnings;
my $FORMAT="%-48s match=%s\n";
while (my $line = <DATA>) {
  chomp $line;

  my $bFound=0;
  while ($line =~ m{"/([^"]+)"|/(\S+)}g) {
    my $sMatch = defined($1) ? $1 : $2;
    print sprintf($FORMAT, "<$line>", "<$sMatch>");
    $bFound=1;
  }

  print sprintf($FORMAT, "<$line>", "---") unless $bFound;
}

__DATA__
"/moreIters 10"
"/blah X Y Z"
/fewIter
/some stuff here
"/albatross" foo bar baz
monkeys
leprechauns /not monkeys
/gnomes "not leprechauns though"
/grumpy gnomes "/silly elves" "/funny" fairies
[download]

which outputs

<"/moreIters 10">                                match=<moreIters 10>
<"/blah X Y Z">                                  match=<blah X Y Z>
</fewIter>                                       match=<fewIter>
</some stuff here>                               match=<some>
<"/albatross" foo bar baz>                       match=<albatross>
<monkeys>                                        match=---
<leprechauns /not monkeys>                       match=<not>
</gnomes "not leprechauns though">               match=<gnomes>
</grumpy gnomes "/silly elves" "/funny" fairies> match=<grumpy>
</grumpy gnomes "/silly elves" "/funny" fairies> match=<silly elves>
</grumpy gnomes "/silly elves" "/funny" fairies> match=<funny>
[download]

Best, beth