Assuming you are referring to an IMAP account, you must be online to access the account. When offline, you can delete what is shown on your PC, but the message remains on the server until you can sync again.

Sorry, David. It's me again. The e-mails are all downloaded to my notebook, not kept on the server. Does it make a difference? On my phone, I can delete messages on a plane and offline, and when I land, these changes are made. I would love to be able to archive the messages and delete them on my notebook when off-line, albeit I fear this may not be possible.

The issue is that the notebook must, at some point, go online so the IMAP account will sync and delete the message from the server. And, if the account is IMAP, it is impossible to download to PC and not be kept on server. Is this a POP account?

Hi, Yes, it is a POP account.

It used to work wonderfully well in the past, I could handle mails on trains, planes and taxis, but since I installed a new version that possibility disappeared. This has reduced the flexibility greatly.

It might be your settings. If thunderbird is set to download and delete, then all will be on your PC, but if the account is set otherwise, you will have messages on both server and PC. Look at the 'server settings' pane for the account.