There is a dress code - preferably shirt and tie, no jeans and no trainers.
All depends on the circumstances. In October of 2001, a friend's eldest son was upgraded on a US domestic flight to first class and I'm sure he wasn't wearing a tie. While they didn't explain why they were upgrading him on a flight that wasn't sold out, we strongly suspect that the fact the he's a big, strapping lad and in the US Army (101 airborne) probably had something to do with it. They put him in row #1 aisle seat and said in no uncertain terms that he was to sit there.