Anyway - LIKE is fail as well

Sorry, but that is not an adequate problem description. Please see How do I post a question effectively? and I know what I mean. Why don't you?, and provide a Short, Self-Contained, Correct Example, like the following. Note how it is exactly the code you posted, except that it works fine for me.

use warnings;
use strict;
use feature 'say';
use DBI;

my $dbh = DBI->connect(
    "DBI:mysql:database=testing;host=127.0.0.1", $ENV{USER},
    'barfoo', { RaiseError => 1, AutoCommit => 1 });
$dbh->do('DROP TABLE IF EXISTS texts');
$dbh->do(<<'ENDSQL');
CREATE TABLE texts (
    string VARCHAR(256) 
);
ENDSQL
$dbh->do('INSERT INTO texts (string) VALUES ("/a/b/c");');

# ---
 my $string = '/a/b/c';
 my $req = qq{ SELECT * FROM `texts` WHERE `string`=?};
 my $sth = $dbh->prepare($req);
 my $row = $sth->execute($string);
 say "row = $row";
# ---

my $sth2 = $dbh->prepare(
    q{ SELECT * FROM texts WHERE string LIKE '%/a/b/c%' });
$sth2->execute;
say "row = $row";
[download]

This prints row = 1 twice, as expected.

This is how I spun up a fresh test MySQL instance. To change the server's port number, change the -p3306:3306 to something like -p13306:3306 in the docker command, add the argument --port=13306 to the mysql command, and add the argument ;port=13306 to the DBI connection string in the Perl code.

$ docker run --rm --name testmysql -p3306:3306 -e MYSQL_ROOT_PASSWORD=foobar -d mysql:5
# wait a few seconds for it to start
$ echo "CREATE USER '$USER' IDENTIFIED BY 'barfoo'; CREATE DATABASE testing; GRANT ALL PRIVILEGES ON testing.* TO '$USER';" | mysql --protocol=TCP --user=root --password=foobar
# after testing:
$ docker stop testmysql

Several smaller edits to add more information.

Sorry for delay. THis is my code

 $dbh->do("CREATE TABLE IF NOT EXISTS `test` ( `string` VARCHAR(256))"
+);

 my $string = '/a/b/c/d';

 my $I = qq{ INSERT INTO `test` (`string`) VALUES (?) }; 
 my $sI = $dbh->prepare($I);
 $sI->execute($string);

 my $R1 =  qq{ SELECT * FROM `test`};
 my $sR1 = $dbh->prepare($R1);
 my $row1 = $sR1->execute();
 say "total1: $row1";
 while( my $L1 = $sR1->fetchrow_hashref )
    {
     say "$L1->{'string'}";
    }

 my $R2 =  qq{ SELECT * FROM `test` WHERE `string`=?};
 my $sR2 = $dbh->prepare($R2);
 my $row2 = $sR2->execute();
 say "total2: $row2";
 while( my $L2 = $sR2->fetchrow_hashref )
        {
         say "$L2->{'string'}";
        }
[download]

produce:

total1: 1
/a/b/c/d
total2: 0E0
[download]

 my $R2 =  qq{ SELECT * FROM `test` WHERE `string`=?};
 my $sR2 = $dbh->prepare($R2);
 my $row2 = $sR2->execute();
[download]

You forgot to pass $string to ->execute as a value for the placeholder. $sR2->execute($string); works as expected.