Pretty simple, actually.
I decided to view the sources of one of these malfunctioning pages, and the last few lines of HTML basically gave away the problem:
So after the large chunk of HTML that makes up the dashboard menu, WordPress is supposed to load up the content that lets you customize your blogs and whatnot. However, it only renders the identifier for it and stops after that.
This can pretty much be a server-side error not caused by WordPress, but most likely by poor management of the internals, either from a tangle in the database or from an error in the core of WordPress itself.
I’m not sure if @MichaelPower will have enough time to look into the situation, also considering that he’s in a pickle with managing Spaces itself. But if he does, then I hope it can be soon enough where it doesn’t cause prolonged problems.