I came across a pretty nifty module that allows small number count icons to be positioned next to menu items and tabs, allowing things like "number of unread private messages" to be placed on an account menu icon like so many social media sites.

The only problem is that Menu Badges (menu_badges) and Menu Icons (menu_icons) are incompatible. The badges don't show up on items with menu icons.

The maintainer of that module mentioned this in this issue queue over there at menu_badges:
https://www.drupal.org/node/1702938
#1702938: menu badges with menu icons (links with icons only, no text)

I took a look at the code for Menu Icons. That module overrides the standard theme_link when the text hiding option is enabled. This makes it very difficult for me to integrate with it. You should open an issue over on the Menu Icons queue. I'm pretty sure the maintainer of that project could use the preprocess technique I use in Link Badges instead, and the two modules would work together.

Any change of making Menu Icon compatible? Menu Badges really makes it quick and easy to implement that functionality.

https://www.drupal.org/project/menu_badges