Beagle - 12:29 am on Feb 4, 2006 (gmt 0)

It's definitely "inquire" and "inquiry" in the U.S., according to me and two dictionaries that I checked just to be sure. ;-) All "enquire" gets is a referral to "inquire."

But I'd say you're correct that "inquiry" has the sound of an investigation to it. It's a very formal sounding word to an American ear, not something to be used in everyday speech. When I read "booking inquiry" I really didn't know what it meant, although I guess it would be asking about registering/booking for something? Unless you have a purpose in coming across in a very formal way, something like "Ask us about..." would be a more American-friendly invitation.