I'm not sure any cartridge ever becomes obsolete except for all rimfire cartridges larger than .22. Likewise, I'm not sure anything old would be an oddball just because it's old but would be only in comparison with contemporary cartridges. Anyhow, I have some favorites.
First, in revolvers, the .38 S&W but not for any particular reason. I see someone already mentioned it, though. More interesting, however, is the .38-40, which I believe was originally called the .38 W.C.F. I think John Wayne used one in his early cowboy pictures.
In a rifle, there was a .32-40 Winchester/Ballard. I never owned one but I once got to live in the same house with one for a long time. It was a Winchester Hi-Wall and it had a long, heavy octagonal barrel. I never saw anyone take it out and shoot it (and there was a lot of that going on around there) because I doubt anyone had any ammo for it. I have no idea where it is today.