Ok, that probably would make more sense if I put a screenshot of it: Appears to be for all the eggs awarded for battling - haven't got any other no-trade items to test it with

I'm assuming this is a bug, as usually the delete link for an item would be on a line of its own, and separated by a line break before you get the other options.... isn't it? Not only is having it on the same line really confusing, but it makes mis-clicks even easier! I guess it's for the eggs awarded for battles, because they don't have any vault / shop options, and of course you can't donate / give it to a buddy.

I don't have any CustomCSS active, so it's not that messing things up.

This may have been like this all the time (and apologies if it's already been mentioned - I did hunt around and didn't see anything else about it) - I've only just clicked on it now to use it on a pet (I've had it sitting in my inventory for months before I had a pet that I wanted to use it on :P), and noticed the delete link is on the same line as the top feed link.