boot - What kernels & packages do I un-install?

Okay, my wife's computer is apparently running out of space on /boot and from what I can tell in similar questions, it's related to unused kernels (which makes no sense, considering it's a fresh installation!)...

But what do I actually un-install?

Other questions here indicated I needed to keep the most recent kernel (linux-image-3.13.0.34-generic?) and the backup/fallback (linux-image-3.13.0.32-generic?)...

So does that mean I should be removing everything else (linux-image-3.13.0.xx-generic, linux-image-extra-3.13.0.xx-gen and linux-image-generic)? Do I need to keep those "linux-image-extra-3.13.0.32-gen" and "linux-image-extra-3.13.0.34-gen" packages, if this is the case?

I was able to find plenty of questions telling me how to remove unused kernels... But not too many telling me how to identify (well, "confirm" really) what has to come off.

Any help in identifying exactly what has to be removed would be appreciated.

First thing to is run:

uname -r

This will tell you which version you are currently using.
Then run

apt-cache policy linux-image-generic

The Installed and Candidate fields should be the same. (If not, you have an upgrade available - make the upgrade). That value is the latest version you have installed.

Removing everything except the latest version and the currently running version is safe¹. So in this case, remove packages which contain all the other versions (both image and image-extra). There may also be header packages that you can remove.

¹ If both are the same, then remove everything but the last two versions - always have an extra version as backup, preferably one you know works well.