There have been many threads I have searched for where users have unanswered issues receiving alerts from osTicket.
I will cover the case for using phpMail with v1.6 ST
First, ensure your php mail settings are correct.
Write the following script to a file (test.php or something) and run it:
<?php
mail('youremail@yourdomain.com','test subject','test body');
?>
Wait a couple minutes. Did you get the email?
If not, first check that your mail provider did not sort it into a "junk" or "spam" folder. If it is still not in your inbox, then re-check your mail configuration and the way it relates to your php config.
NOTE: If you are sending the mail to an address that is on the same domain as your server but your server is not the mail server for that domain, then you will need to change your mail configuration (sendmail, postfix, etc) to disable local delivery. Here is link for sendmail: (http://serverfault.com/questions/65365/disable-local-delivery-in-sendmail) I use postfix and simply changed the 'mydestination' param in '/etc/postfix/main.cf' to remove that host from the list.
If you got it, but still do not get email notifications from tickets, next check your osTicket Alert Settings.
From admin page, click 'Settings' then scroll down to 'Alerts & Notices'. Make sure these settings are checked appropriately for what you require.
After verifying 1) your phpmail config and 2) your osTicket settings, you should be receiving alert emails from osTicket
If there are other things to check regarding mail config for osTicket, please post them here.