A closing choice on the Scedule page of a ticket is the option to delete this ticket.  Selecting this option will first move the ticket to the recycle bin.  Once it is in the recycle bin you have the choice to permanently delete this ticket or remove from the recycle bin.  Only a user with Level 5 access can empty the Recycle Bin.

On the ticket List page a closing choice is the abilty to batch delete.  You also have the ability to batch remove from the deleted file.

Hint:  You can delete a ticket without deleting the Contact/Customer.  You cannot delete a Contact/Customer without deleting all tickets tied to that Contact/Customer.