TB doesn't appear in 'Get notifications from these senders' because currently there's no integration with the W10 Notification Center. If you turn TB ON in Personalization/Taskbar/Select which icons appear in the taskbar, then you will see the envelope icon in the Notification Area when TB is open and new mail arrives, and 'Show a tray icon' is checked in TB Tools/Options/General. To have a sound, check the 'Play a sound' box in General and choose a sound file.

Consider also this add-on: https://addons.thunderbird.net/en-US/thunderbird/addon/unread-badge/

Many thanks. I had those settings as you describe, using "Default system sound for new mail". That's the one that doesn't work; it doesn't play the appropriate system sound. As a workaround I pointed the "Use the following sound file" to one of the system sound files, and that seems to work. So I guess there must be something odd about my setup; perhaps at some stage I should try a clean reinstall of TB and see if that makes a difference.

Click the Start button, type sounds, click Change system sounds, Control Panel, and make sure the Sound Scheme is not No Sounds (see picture). That should enable the Default system sound in TB.

Very strange. That's how I had things set up. So I did a bit of scrolling around in the Sound listing and found another one I'd not noticed before (see below) called "Desktop Mail Notification". That's the one my installation of TB seems to be using as the default system sound for new mail, not the "New Mail Notification".

Thanks again for your help.