Any saved search can be deleted. To delete a search, click on the Search button to see a list of searches. Double click the search you wish to delete. Click on the details link to the top of your screen. This will open a new panel to which you will need to locate the 'Delete search' button in the upper right.
Click this button, you will need to confirm your actions.
Once a search has been deleted, it is gone permanently. It cannot be retrieved. the good news is that all is not lost, if you knew what the criteria was for the search, simply create a new search.
When deleting searches, all that you are deleting is the search criteria, the data itself remains in the archive.