The Microsoft Update Catalog web site seems to be very unreliable. If it isn't working, what troubleshooting steps are sometimes helpful?
(Please note that the Microsoft Update Catalog is not the same thing as Microsoft Update, although they offer the same content. The Microsoft Update Catalog is a web site where you can download off-line installers, including installers for different versions of Windows than the one you are running. It requires the use of Internet Explorer, and is surprisingly fragile.)
Note: this is intended as a canonical question, to collect various possible solutions to the various common problems with the Microsoft Update Catalog. I'm posting an initial answer, but additional answers are encouraged. Posted to Server Fault rather than Super User because the MUC is primarily used by system administrators, end users will typically use Windows Update or Microsoft Update instead.