It appears that site registration is mostly broken.
edit: Partially resolved. The system now proceeds to the page that says the activation email has been sent, even though activation is not needed. You can click Log in and proceed from there.
edit2: Resolved! There's a "redirect" option in the plugin, which I sent back to the main page. The original problem turned out to be with the Gmail SMTP plugin I was using for outbound e-mail. Turns out something had happened, and it was failing authentication. I disabled it (for now), so site-wide e-mail is currently not working. Any password resets will need done manually until I get it working again ;)
edit3: site-wide email should work again as well :)