UPDATE 28 OCT: I just updated to Civi 5.6 and PHP 7.1 and this is still a problem. Grateful for any other suggestions.
I'm having pretty much the same issue as this person did not long ago.
I'm unable to add event registrations for contacts through the "Add New Registration" button. The drop-down menu doesn't appear, although oddly if I close the "New Event Registration for" pop-up I can see the drop-down menu 'floating' awkwardly on the page. Looks like a script error. This is affecting the 'event', 'participant role' and 'participant status' options. (Unlike the other person with this issue, my 'accordion' options on the record are appearing just fine.)
And here's the weird 'floating' menu which appears if I try to type something then close the "New Event Registration for" modal:
I've checked my custom data type fields as suggested in that link, but are all as they should be.
What else can I try?
I'm using Wordpress with Civi 5.4 (Civi 5.6 as of 1 Oct).