This post is not meant to be a how to, but rather a compilation commands that could be helpful in trying to fix issues with the login screen after an upgrade.
These commands should be executed from a prompt, you can get to one using the ctrl+alt+f1 (you can replace f1 with any of the function keys from 1-7), or you can boot up from recovery, or you could mount the drive and chroot into it.
Reinstall Desktop and Login Manager
sudo apt-get install --reinstall lightdm ubuntu-desktop
If your using using Gnome Shell
sudo apt-get install --reinstall gdm3 gnome-shell
Reconfigure lightdm, set it as the default login manager
sudo dpkg-reconfigure lightdm
Replace GDM with LightDM
sudo apt-get remove gdm3 sudo apt-get install lightdm
Installing GDM (Gnome Login Manager)
sudo apt-get install gdm3
Remove Intel video and Plymouth
sudo apt-get remove xserver-xorg-video-intel plymouth
Reboot, get to a prompt and reinstall login manager, replace lightdm with whatever one your using
sudo apt-get install --reinstall lightdm
Reinstall Intel video and Plymouth
sudo apt-get install xserver-xorg-video-intel plymouth
Install Ubuntu Gnome Desktop
This seems to solve some of the issues with GDM3 not starting up.
sudo apt-get install ubuntu-gnome-desktop