And where do you find them in the webmail interface? In which folder? Is that folder subscribed in Thunderbird?

In the Inbox, and yes.

Are you sure that they are disappearing because Thunderbird wrongly thinks they are junk, or are they just disappearing? If the latter is a possibility you may find this article helpful: http://kb.mozillazine.org/Thunderbird_:_Issues_:_Disappearing_mail

Nope, none of the above, but thanks.

The app sometimes flashes a "moving message to trash" note, but the message does not appear in the trash. It is gone completely.

But if they are still in the Inbox they haven't been deleted…

They are not in my local Thunderbird's inbox, they are in my Webmail inbox.

Well, if your account uses POP I'd expect it to delete the messages from the server after downloading to Thunderbird, unless you have changed the default setting in order to make it leave copies on the server.

The more general case these days is to use IMAP, where the mail client mirrors what is on the server and vice versa. In this case, if it is being deleted from the client (Thunderbird) then by right it ought to be deleted from the server too. That's what IMAP is all about.

So, a message being deleted from Thunderbird and yet remaining on the server gives us something of a mystery.

I wonder if Thunderbird has become set to show only unread messages. But then there is the strange "moving message to trash" message.

Wanna know where I found them? in TBird, in the Trash, in the Local Folders section. Which I do not use any more because it was omitted from a backup I really needed one time.