On lines 905 and 906 of comment-template.php, at the top of the comments_popup_link function you will find this:
if ( is_single() || is_page() ) return;
It makes it so that you can't list the comment count of an article if it's on a single or page. This unfortunately includes sidebar widgets and, in my case; a feature box that appears on the homepage, which is a static page.
I can't think of any conceivable reason for this code to be here; I commented it out myself and everything still works the same. I'd rather not have to re-comment it every time wordpress updates though.
Please consider removing it because, as far as I can see, it just doesn't have any useful reason to be there and only gets in the way of theme designers like myself.