I'm having a strange issue - No matter with what user I login in my front-end Joomla site (even administrator) and try to add an article and then press button 'publishing', I do not see any category to choose from, it is all blank.
As shown in the screenshot, it's not only the category drop-down, but all the drop-down elements (status, featured, access etc).
I would guess that this might be a CSS issue with your template. The only way to tell is by checking with the dev tools in your browser. Right click on the drop-down area- choose "Inspect Element" in and check for the following:
One quick way to see if it's Template related, you can change the front-end template - switch it to the default Protostar template and try again.
If the site is live and you don't want to make this change visible to everybody, create a new test article - assign it to a hidden menu item, where only you will know, then assign to this menu item the Protostar template. and navigate there in the front-end and try to edit it.
If you find out that it's the template that causes this, contact the template developer regarding this issue.