The key is to accumulate your string as you loop through the directories. Not a one-liner, but this split solution gets the job done...

use strict;
use warnings;

my $s = '/home/code/work';
my @ds = split m{/}, $s;
shift @ds;
my $accum;
for (@ds) { 
    $accum .= "/$_";
    print "$accum ";
}
print "\n";
[download]

Update: or, adapting your regex attempt:

my $s = '/home/code/work';
my $acc;
while ($s =~ m#([^/]+)#g) {
    $acc .= "/$1";
    print "$acc "
}
[download]