
My indoor club runs the same method as Andy632's. The only flaw in the system is when you acquire a new firearm between membership renewals the new firearm doesn't have a number so I'm not entirely sure what you're supposed to do. Should be have a new rifle by next week i'll be using there so i'll ask if it can just be added to my records I suppose.
Other clubs I visit, as a guest, require you to write out your serial number(s) when you sign in, which can get a bit tedious, especially if you're their with a few guns. As such I've geekily printed off a ton of small sticky labels with the serial number of each gun on which cuts down the hassle. The only difficulty is peeling the bloody things as they're really small!