Thanks to Jez for pointing out that since I rebuilt my server over the New Year period, the submit buttons on my sites haven't been working on certain versions of Internet Explorer. Turns out it was entirely due to my being a good boy and installing Service Pack 1 for the .NET Framework, as this thread from Paul Wilson's blog explains:

The problem is resolved by copying the newer version of WebUIValidation.js into the aspnet_client folders for each website. There's one to remember for the future.

Upside:  Everything still worked fine in Firefox and other "down-level" (arf!) browsers which don't benefit from the WebUIValidation.js code.

Downside: Everything also still worked fine for the evil comment spammers who POST directly to the relevant URL and bypass the form and it's submit button altogether. So I'm good for online poker and viagra links for a while, thanks.