> but it is intended to use the last successfully matched regexp

Whatever its intent, it is one confusing puppy. //; always matches, always returns TRUE, but it never changes $&. $& is always whatever the previous regex set it to, whether it matched or not, effectively a NOP. Consider this:

$_ = 'Hello Perl';
say '$_ = \'Hello Perl\';';
print "\$1: $1\n\$2: $2\n\$3: $3\n\$&: $&\n\n";

# unsuccessful match
/Python/;
print "No match, \/Python\/\n";
print "\$1: $1\n\$2: $2\n\$3: $3\n\$&: $&\n\n";

# successful match
if (//) {
    print "No nothing, if (\/\/) {\n";
    print "\$1: $1\n\$2: $2\n\$3: $3\n\$&: $&\n\n";
} else {
    print "\/\/ unsuccessfull match";
}
 
# successful match, no captures
/Perl/;
print "Match \/Perl\/, No captures\n";
print "\$1: $1\n\$2: $2\n\$3: $3\n\$&: $&\n\n";

# successful match, empty pattern
if (//) {
    print "No nothing, if (\/\/) {\n";
    print "\$1: $1\n\$2: $2\n\$3: $3\n\$&: $&\n\n";
} else {
    print "\/\/ unsuccessfull match";
}
 
# successful match, no captures
/Perl/;
print "Match \/Perl\/, No captures\n";
print "\$1: $1\n\$2: $2\n\$3: $3\n\$&: $&\n\n";

# successful match, no pattern, empty parens
if (/()/) {
#//;
    print "No nothing, if \(\/\(\)\/\) {\n";
    print "\$1: $1\n\$2: $2\n\$3: $3\n\$&: $&\n\n";
} else {
    print "\/\/ unsuccessfull match";
}
[download]

Which results in:

  $_ = 'Hello Perl';
  $1: 
  $2: 
  $3: 
  $&: 

  No match, /Python/
  $1:  
  $2: 
  $3: 
  $&: 

  No nothing, if (//) {
  $1: 
  $2: 
  $3: 
  $&: 

  Match /Perl/, No captures
  $1: 
  $2: 
  $3: 
  $&: Perl

  No nothing, if (//) {
  $1: 
  $2: 
  $3: 
  $&: Perl

  Match /Perl/, No captures
  $1: 
  $2: 
  $3: 
  $&: Perl
  
  No nothing, if (/()/) {
  $1: 
  $2: 
  $3: 
  $&: 

What is or was the purpose of this construction? How would one use it?

> What is or was the purpose of this construction? How would one use it?

#!/usr/bin/perl
use warnings;
use strict;
use feature qw{ say };

$_ = 'abacad';

say "/a(.)/";
if (/a(.)/g) {
    say "\$1: $1";
    say "\$&: $&";
} else {
    say 'No match';
}

for my $try (1 .. 3) {
    say "//";
    if (//g) {
        say "\$1: $1";
        say "\$&: $&";
    } else {
        say 'No match';
    }
}
[download]

Output:

/a(.)/
$1: b
$&: ab
//
$1: c
$&: ac
//
$1: d
$&: ad
//
No match
[download]

Update: If I remember correctly, this was the original reason the feature was introduced:

#!/usr/bin/perl
use warnings;
use strict;
use feature qw{ say };

my $x = 'found 11';
my $y = 'found 12';

if ($x =~ /found (\d+)/ && $y =~ //) {  # No need to repeat the long r
+egex! Yay!
    say "Found $1.";
}
[download]

map{substr$_->[0],$_->[1]||0,1}[\*||{},3],[[]],[ref qr-1,-,-1],[{}],[sub{}^*ARGV,3]

I changed my mind about sitting this one out because this code from choroba is so fascinating, and worth exploring, as noted below.

#!/usr/bin/env -S perl
 
#use warnings;      # suppress uninitialized warnings
use strict;
use feature qw{ say };
use experimental qw( signatures );


$_ = 'abXXXXacVVVVVad';
#$_ = 'abacad';

my $lim = 7;
say "Examining \'$_\' $lim times";
say "/a(.)/";
if (/a(.)/g) {

#say "/q(.)/";          # switch to see no-match example
#if (/q(.)/g) {

    say "\$1: $1";
    say "\$&: $&";
} else {
    say 'No match';
}

for my $try (1 .. $lim) {
    say "//";
    if (//g) {
        say "\$1: $1";
        say "\$&: $&";
    } else {
        say 'No match';
    }
}

$_ = '37.5BBBBB98UUUUU4.075QQQQQ42TTTT0.357SSS';
$lim = 5;
say "\nExamining \'$_\' $lim times";
say '/[^\d.](\d+(?:\.?\d*)?)/g';
if (/[^\d.](\d+(?:\.?\d*)?)/g) {
    say "\$1: $1";
    say "\$&: $&";
} else {
    say 'No match';
}

for my $try (1..$lim) {
    say "//";
    if (//g) {
        say "\$1: $1";
        say "\$&: $&";
    } else {
        say 'No match';
    }
}

exit(0);
__END__
[download]

  O U T P U T

  Examining 'abXXXXacVVVVVad' 7 times
  /a(.)/
  $1: b 
  $&: ab
  //
  $1: c
  $&: ac
  //
  $1: d
  $&: ad
  //
  No match
  //
  $1: b
  $&: ab
  //
  $1: c
  $&: ac
  //
  $1: d
  $&: ad
  //
  No match

  Examining '37.5BBBBB98UUUUU4.075QQQQQ42TTTT0.357SSS' 5 times
  /^\d.(\d+(?:\.?\d*)?)/g
  $1: 98
  $&: B98
  //
  $1: 4.075
  $&: U4.075
  //
  $1: 42
  $&: Q42
  //
  $1: 0.357
  $&: T0.357
  //
  No match
  //
  $1: 98
  $&: B98

S O M E  O B S E R V A T I O N S

The only use case I can infer from these code tweaks is the notion of determining the number of 'things' present in a string, as long as at least one of those 'things' is present, since there must be at least one match before //g can work its magic. I deliberately omitted a leading non-decimal in the second $_ to force the first match deeper in the string, but that's cool because as long as at least one 'thing' sub-expression is present, //g will find the rest (to the right). Also, note how the entire process starts over from the beginning if the number of $trys exceeds the number of 'things' present, as in $try (1..$lim) {. There must be other use cases for //g;. Any thoughts on how to exploit this facility?

Will

// always matches, always returns TRUE, but it never changes $&.

That is not correct, in either aspect:

% perl -wle '"a" =~ /a/; "b" =~ // or print "did not match, did not re
+turn TRUE"'
did not match, did not return TRUE
% perl -wle '"a" =~ /.*/; q{$& changed} =~ // and print $&'
$& changed
%
[download]

I concede, I give up and I surrender. Too confusing, why I get the result I do and you get the results you do. I'll sit this one out and just study the results y'all get. :-)