I haven't gotten any answers (or even any views of my question), so here's my very non-authoritative answer, based on some googling and some testing of one of the programs.
There's a comprehensive list of NoSQL databases here: http://nosql-database.org/
So you can search that web page for ".net" or "c#", and get a complete (and quite large) list of possibilities. And don't forget to click on "A NoSQL parody" over in the right hand column.
For large systems two leading contenders are these:
mongoDB: https://www.mongodb.com/ https://docs.mongodb.com/ecosystem/drivers/csharp/
For medium-sized systems this one seems to be popular:
For small, perhaps serverless, systems there's these two:
All of the above are written in C# except for mongoDB, which has a "C# driver".
After a bit of testing I've chosen LiteDB for my program. It seems to be well-maintained and has quite a bit of usage - it's been forked almost 500 times on GitHub.
(I also asked if it was a good idea to store serialized .Net objects in a database. I'm still very interested in possible answers / opinions on that question.)