If you're using Intel graphics then I don't think this is going to be a driver issue.
My first guess is that a gnome-shell extension has broken rendering for the panel menu. Please uninstall (not just disable) all non-standard* gnome-shell extensions and tell us if the problem persists without extensions installed.
* non-standard means everything other than gnome-shell-extension-appindicator and gnome-shell-extension-ubuntu-dock.
If you're using Intel graphics then I don't think this is going to be a driver issue.
My first guess is that a gnome-shell extension has broken rendering for the panel menu. Please uninstall (not just disable) all non-standard* gnome-shell extensions and tell us if the problem persists without extensions installed.
* non-standard means everything other than gnome-shell- extension- appindicator and gnome-shell- extension- ubuntu- dock.