System won't boot anymore after software update

I think grub failed to update properly after the apt updates. Now that boot repair can't fix it for you, you need to fix it manually. Your boot-repair info shows three drives: sda, sdb and sdc. You need to find out where you installed ubuntu. If it's sda, be sure to use that for all the commands mentioned below.

1. Boot your computer from the Ubuntu DVD (of from a Live USB with Ubuntu on it). Select "Try Ubuntu (without installing)". Only use the DVD of the Ubuntu version that you wish to repair. So for 15.10, use the DVD of 15.10.
2. Now you need to find out what's the name (and the partition number) of the root partition (the partition on which your Ubuntu has been installed). Check this with the application Gparted Partition Editor. Gparted is present both on the Ubuntu DVD and on the Linux Mint DVD.

Tip: the root partition of your Ubuntu will usually be formatted in EXT4. The EFI partition wil probably be formatted in FAT32, and it'll have the boot flag. On my computer, the Ubuntu root partition is called sda5. For the sake of clarity I'll use that situation in the rest of the how-to.

3. Launch a terminal window.

4. Copy/paste the following command line into the terminal (you can copy/paste with a right-click of the mouse): sudo mount /dev/sda5 /mnt Note: only if Ubuntu is on sda5! Change the number according to your situation. ....and press Enter. This mounts the Ubuntu root partition on the hard drive.

5. Copy/paste this command line into the terminal: sudo grub-install /dev/sda --root-directory=/mnt

....and press Enter. This installs Grub. Note: definitely use copy/paste and do not type this command by hand! Many people forget to type essential spaces in this command. Note: you might get the following harmless error report: grub probe: error: failed to get canonical path of /cow Installation finished. No error reported. This grub probe error is harmless, and all should be well.

6. Now type in the terminal: sudo reboot ....and press Enter.

This should work! You will find more grub related info on this link.