If you, like me, have an old setup that used to work but the tools updated in the meantime and some things are just not the way they used to be, you might find this useful.
It seems that in the newest version of Omnisharp (at least the one I am running) “Omnisharp: Use Global Mono” option does not exist anymore.
When I booted new Unity project for this course, my VSCode did not want to start Omnisharp at all. What finally helped was finding “Omnisharp: Use Modern Net” option in the settings. It seems to be set to true by default but (as the helptext helpfully points out) “this version does not support […] including Unity”.
Turning this off (and later updating dotnet from 5.0 to 6.0 when it complained) and restarting the editor fixed the issue.
Hope it helps someone.