Problem:
An Ubuntu Server system has been upgraded with do-release-upgrade from 14.04 to 16.04. SSH connections are possible and working fine, but there is no output (or only a fsck message) on the local console (for example VNC console at your VPS provider). Login via the local console / VNC console is therefore not possible, while connections via SSH are possible.

Solution:
As a temporary workaround, you can press Ctrl + Alt + F1 to get to the login prompt. Alternatively, connect to the system via SSH.

To fix the issue:
Open /etc/default/grub in a text editor (for example, nano):

sudo nano /etc/default/grub

Change the option GRUB_CMDLINE_LINUX_DEFAULT from splash quiet to text quiet

Then run sudo update-grub.
After a reboot, the issue should be resolved.