Broken 14.04 LTS while upgrading to 16.04

Solution 1:

You could try booting into an older kernel from the grub menu. Holding the shift key while booting should give you the grub boot menu.

Hopefully you will see a newer kernel that was installed as part of the upgrade and also an older kernel version to boot into.

If the older kernel is an option then boot into it. Then you can try to fix any broken packages with

sudo apt-get -f install

From previous experience I normally give up and do a clean install if things don't go smoothly when upgrading. In fact when I went from 14.04 to 16.04 I just choose the clean install option. You may be able to perceive and fix the boot problem only to find other issues exist and you may spend more time than its worth fixing them.

Worst case (and probably easiest option) I would suggest using a live cd to boot and get all your files off onto a USB drive then clean install.