I've used Events Manager once before (for an event in October 2012), and it went swimmingly.
I created a new event, yesterday and made the mistake of not testing it before sending out the link. When users enter their data and click "Register," the form momentarily changes to the spinning wheel, but does not do anything else.
I've tested the past event (that did work) and the same issue occurs, now. On my dev site, I've disabled all other plugins, reinstalled Events Manager 5.3.5, resolved all JS console errors (Jetpack stats) and even switched to the TwentyTwelve theme, without it having any effect on the plugin functionality.