When I enable "Search Engine Friendly URLs" and then click on any events register button, the page is displayed without any styling, like the path to the css file was not found.
When I toggle SEF urls off, all returns to normal.
Are you using a 3rd party SEF component like sh40SEF or anything else? If so, you must set them to SKIP DT Register. However, you should have no problems with DT Register just using the built-in core SEF of Joomla. If that's all you are using, submit a ticket so we can check it out.