For anybody interested, what I did now was force Grub2 to use a higher frame buffer resolution.
Open the grub-settings file with
$ sudo gedit /etc/default/grub
uncomment the line #GRUB_GFXMODE=640x480 and modify it to
GRUB_GFXMODE=1920x1080
and potentially add
GRUB_GFXPAYLOAD_LINUX=keep
Then run
$ sudo update-grub
$ sudo reboot
For me it would push up the resolution to full-hd but you could see sort of scan-lines when switching a window, but maybe it works for you. I then simply used the VGA resolution option instead as proposed on this Ubuntu bug report by commenting again the inserted lines above and uncommenting the line GRUB_CMDLINE_LINUX_DEFAULT="quiet splash" and modifying it to
GRUB_CMDLINE_LINUX_DEFAULT="quiet splash vga=792"
where the code 792 corresponds to a resolution of 1024×768. I did not have luck with higher resolution codes such as 1600x1200, they again would suffer from the same bug.
Run
$ sudo update-grub
and restart again and you should have at least a system with a 1024x768 resolution.