Thanks to Jake_ on the IRC for getting me to look into this.
Chrome 45.0.2454.85 m: Works on all except XP.
Firefox 40.0.3: Works. Even on XP.
Internet Explorer: 8 (XP) unsurprisingly fails; 11 works on Windows 7 and 8.1
(Note: "XP" means the 32 bit variety. I consider Windows XP x64 pretty rare, and don't have a copy handy to test on, so I don't know.)
I'm posting this as a separate thread for now so it gets seen, but it'll probably get merged into the "SSL hiccups" thread later.
Technical aside: The way MyBB works, we can either put http:// or https:// on its idea of the board's URL. Whichever we put, it will write that in the URLs to load all the assets for a page. So, we have a choice between giving HTTPS users pure-HTTPS pageloads, and letting HTTP users load everything over plain old HTTP. We chose the former. This shouldn't be a problem, right?
...Of course it is! Some older browsers and OSes can't deal with the particular variety of HTTPS our server uses. Following reasonable security guidelines, our server supports TLS 1.0, 1.1 and 1.2, and we have an ECDSA certificate, so all its supported cipher suites use ECDSA signing. Unfortunately, some browsers can't handle that -- including the latest Chrome, which can only handle RSA certs under XP.