A search for "invite.ics" works fine here.

Thanks for the suggestion to search for "invite.ics" in attachments. This works for some invitations (in my case, actually older invitations). However, this search does not find plenty of invitations which do not show an attachment at all (kind of "inline" data?), e.g. sent by Outlook, see uploaded screenshot. This example invitation, opened as plain source, contains several "Content-Type" lines, i.a. "Content-Type: multipart/alternative; boundary= ...

.... 

Content-Type: text/plain; ...".

I'd be very happy to be able to search for invitations independent from the sender software/format (or to be sure it is definitely a limited set of e.g. 2 searches that find all invitations).

This add-on https://addons.thunderbird.net/en-GB/thunderbird/addon/expression-search-ng/ should let you search headers.

Great! I can search for from, to etc. fields, as nicely explained in the help lines, but I cannot get the header search to work. May I ask you to type an example, how to search for sth. in the header?

Update: I - almost - found a TB-native way:

• edit / find... / search messages => subject dropdown field, where you can select subject, from, to..., scroll to the very bottom to "Customize"
• in the popup, create a new type called e.g. "Content-Type"
• in edit / find... / search messages => now select "Content-Type" (instead of "subject", "from"...), "begins with", "text/calendar"

This should work both with "inline" and "attachment" invitations, at least the types I find in my inbox - both contain "Content-Type: text/calendar".

HOWEVER, this seems to search only for the first occurrence of "Content-Type:", whereas the "Content-Type: text/calendar" is sometimes not the first and only header entry of this type (in case of a "Content-Type: multipart/mixed" (firt entry of this type) message, where the "text/calendar" part is only e.g. 4th "Content-Type" entry, and thus NOT found.

Any ideas?

I am still interested in Search Expression NG, see post above! Would Search Expressions NG search for all occirrences of Content-Type?

Great! I can search for from, to etc. fields, as nicely explained in the help lines, but I cannot get the header search to work. May I ask you to type an example, how to search for sth. in the header?

headerre:content-type=/mixed/

should work for messages that contain "mixed" in the Content-Type header, but it doesn't return any matches, so forget about the add-on. On the other hand, the native search returns matches, but only if the top-level Content-Type header contains the search terms.

I don't know of anything else to do in Thunderbird. However, if you were to save all messages as text files or eml files, you could run a batch script that searches for all occurences of text/calendar in those files and moves them to another folder. You can them import those messages (eml files only) back into Thunderbird if you want.