I travel with BA on a fairly frequent basis and have Silver status. This allows me to check-in at the club World/Europe desks. I do not always travel club and for leisure I am always at the back. However, I seem to get many more upgrades when checking in at the Club desks. Once my wife and I even got bumped from World Traveller (economy) to Club World (business) on a flight LHR-JFK after checking in with the business guys.
The question I have is: Do the Check-in staff at the business class desks have more leeway in handing out upgrades? No need to focus on BA that was just where I had my experience.