Notifications are sent out based upon the Notification Method set for each Contact.
In order to disable any notifications from being sent out, select the Administration tab, expand Notifications, and select Notification Methods.
Then select a Notification Method which you would like to disable.
Select Edit when the Notification Method window opens and set the Record Status to Inactive and delete the Method in the Notification Method text box.
Save the changes and now this Notification Method will not send out any Notifications.
Repeat the same process on any other Method to disable them.