We were recently on the 26th May crossing from Southampton to New York. There wasn't very much evidence of shorts and ragged jeans at any time. The thing that disappointed my wife and I the most was the lack of any attempt to enforce the formal nights' dress code. I don't suppose the staff can stop people turning up in less than formal gear or refuse them entry, but I think that they could do more to emphasise earlier, maybe in the daily programme, that it is a formal night - I know the daily programme states the dress code, but it isn't really emphasised. Another minor disappointment was the passengers who had chosen to eat in the Lido in the evening who had not bothered to wear a tie or jacket.
I didn't notice any particular nationality being smarter or more casual than others: the two tables of six near to us were all Americans; one table always dressed very formally and smartly; the other lot hardly ever bothered (I think that thay were in a party from Texas - is it generally more casual there?).
We enjoyed dressing up regardless of what anyone else did. I would say that it about 75%+ stuck to the dress code.