one

my @string = grep length, split '\/', 'Call/me/Ishmael/';
for my $i( 0 .. $#string ){
    print join('/', @string[0..$i]),"\n";
}
__END__
Call
Call/me
Call/me/Ishmael
[download]

two

local $fungus = '';
local $_ = 'Call/me/Ishmael/';
s'
 (\W*)?
 (\w+)?
 '
   $fungus .= $1 if $1;
   $fungus .= $2 if $2;
   $2 ? "$fungus\n" : "";
 'gex;
print 
__END__
Call
Call/me
Call/me/Ishmael
[download]

update: nice one borisz++ (i thought about using pos for three, but figured nah)