Mozilla Thunderbird users may have noticed that turning off “Show an alert” for when new messages arrive, does not get rid of the alert that appears in the Windows system tray.
There’s actually a Windows feature (I don’t know if it’s on anything other than XP), that lets you to disable that alert.
On the very edge of the system tray, right-click and select “Customize Notifications” in the content menu.
Select the Thunderbird item, and using the drop-down menu, choose “Always Hide“.
Update: Another method to get to “Customize Notifications” is via the Windows Control Panel -> Taskbar and Start Menu. Select the “Taskbar” tab, and under “Notification area”, click on “Customize”.