I'm not familiar with this particular use of the terminology.

Thunderbird has S/MIME support built in. You can add PGP support via the Enigmail add-on. Both offer the means to sign and encrypt messages.

I see add-ons to support the checking of DKIM-signed incoming messages, but haven't found any yet for signing.

Is it something that an email client is entitled to do? The Wikipædia entry says

Both modules, signing and verifying, are usually part of a mail transfer agent (MTA).

and that means a server, as opposed to a client (MUA).

Yes, usually the server creates a DKIM for outgoing messages.

But maybe there is a plugin for Thunderbird, which can sign letters, if the server does not support DKIM?

Thunderbird can check DKIM signature with "DKIM Verifier" plugin.

As I understand it, DKIM is used to certify that the e-mail really originated in the domain that it claims to come from, thus you'd need to be the owner of that domain and make the respective entries in the DNS registration for this to work. Verifying DKIM at the recipients client on the other hand is straight-forward as long as DNS is accessible to obtain the key, thus can be performed by the client.

I can create a DKIM public key in the DNS, so that email recipients can verify the DKIM signature of letters.

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Many paid email servers (hosting) do not support DKIM.

Maybe there is a plugin for Thunderbird, which can sign letters, if the server does not support DKIM?

I've searched around a bit but didn't find anything either in terms of an add-on that can create DKIM signatures. You could check with the author of the DKIM Verifier add-on if he has some (possibly not public) add-on for testing, or would be willing to help with developing one.

Another option would be to set up a Postfix or similar server locally on your end which adds the DKIM signature and either relays the message to the provider's MSA or acts as the initial MTA itself if the provider hasn't blocked port 25.