When using the default Joomla 3 search module and the button is turned off a 0 appears near the search box.
Possibly more of a workaround until Joomla fixes the code, the solution is to ‘comment out’ the offending line of code when your website does not use the button and is displaying a 0.
Using your prefered file method, such as the cpanel or FTP, open modules/mod_search/mod_search.php
Fine the line that looks like this: (should bearound line 35)
$button = $params->get('button', 0);
and change to:
//$button = $params->get('button', 0);
Now save the changes, clear the Joomla cache (if you use cache) and the 0 has gone.