I actually have no idea where to begin troubleshooting this, since I haven't really touched anything with Magento's backend. The only sin I've committed is by editing the database directly, but only as it relates to attributes. I use Magmi to import products, and have been for a while, and it was after a recent import that I noticed the Category Design tabs were missing. But I am constantly tinkering with this installation, preparing it for launch so it could have been anything.
Disabling modules has no effect. Any ideas where I can start troubleshooting/fixing this, other than reinstalling Magento?
The only tabs I have when editing a category are "General Information" and "Category Products"