How do I disable notification emails in Magento?

We can disable all email communication in Magento by

• Login to Magento backend
• Go to the System->Configuration menu.
• On the left hand menu under the Advanced tab (at the bottom) select the System menu item.
• Select ‘Mail Sending Settings’ from the selection on the right hand side.
• Set Disable email communications to ‘Yes’, then no email will send through Magento.

There is no buit-in ability to disable particular email notifications in Magento. I suggest you to look at the Mage_Core_Model_Email_Template::sendTransactional method. You can overload this class and add check against allowed/disallowed email notifications (based on template name).

Update: you can use extension for disabling specific email notifications in Magento 1.

Also, there is Disable Emails extension for Magento 2.