Generally: reconsider TimelessTemplate::getCategories. What is the point of this, really?

I don't think it would be that good to remove the categories from right sidebar, as they provide valuable information and navigation utitlity in my opinion that is moved into periphery by the standard skins. While it's true, that some pages have a huge category count, this is only true for a minority of them. Furthermore, the right sidebar doesn't need the space for anything else, if you remove the cats you just get whitespace instead.

That's why I think you should go with the second option, but choose a quite high threshold like 20 or anything like that. If you want to select some specific categories – I really don't know how you would do that – you should provide a link All Categories at the bottom of the category sidebar section.

Nevertheless, I think your third and fourth are really valid, but I wonder if you really should remove hidden cats from the sidebar if there's only a small number of categories on that page.

Change 442021 had a related patch set uploaded (by Evad37; owner: Evad37):
[mediawiki/skins/Timeless@master] Fix display of categories in the sidebar

https://gerrit.wikimedia.org/r/442021

Change 442021 merged by jenkins-bot:
[mediawiki/skins/Timeless@master] Fix display of categories in the sidebar

https://gerrit.wikimedia.org/r/442021

Can we use cirrussearch to determine the most important categories to include in the sidebar?

categories are often not things readers would even want to see.

I do not share this vision. Often, the exploration of the content is going through the categories. In Wiktionaries, categories are used for domains and lexicons (for example: words about geometry, names of winds, and so on). We never made any UX test on this, but I think it is something that need to be set visible.

On this matter, I may have missed a dedicated task, but it could be very useful to see the list of categories in preview mode. As a contributor, I miss it a lot!