Hey @StephaneQ
What version of Local are you running? Local by Flywheel version 3 doesn’t play nicely with Catalina due to Virtualbox.
If you haven’t already, can you give Local 5 aka Lightning a try?
You can download it here.
It doesn’t use Virtualbox and works great on Catalina.
You will need to move your sites over to the newer version, here is a guide on that: https://getflywheel.com/wordpress-support/migrating-from-local-by-flywheel-to-local/