How can I identify the latest release in Mercurial?

By looking for 1.9.x?

So today it would be this one?

https://hg.orthanc-server.com/orthanc/rev/409cba9c1dba

The latest version of Orthanc is clearly indicated on the Web site:
https://www.orthanc-server.com/download.php

You can also found the official announcements for new releases in a dedicated thread on this forum:
https://groups.google.com/g/orthanc-users/c/LBil4zk8ZiY/m/99Eoy9dmAQAJ

Regarding Mercurial, check out the following page for the list of the branches (each release has its own branch):
https://hg.orthanc-server.com/orthanc/branches

You can also run the following commands from the command line:

$ cd /tmp/

$ hg clone https://hg.orthanc-server.com/orthanc/

$ cd /tmp/orthanc/

$ hg branches -c

Sébastien-

Merci Sébastien! for your detailed response. Bonne Journée.