Q: I have a 1924 vintage girdle in a box and it is in good condition. The only problem is that some sort of green residue has built up around the metal fasteners and has spotted some of the fabric.
A: The green residue is a chemical reaction that occurs over time from the metal fasteners of the girdle. Just as iron can rust and leaves a brown mark, this metal corrodes to green. Because it's green, it makes me think there is some copper in the metal, but it could be some other metal's reaction.
We see this kind of "dreaded greenie" on vintage jewelry, as well as purse hardware and sometimes on girdles like you're describing. The easiest remedy for the hardware itself is to rub it away with an ear swab. Sometimes this works better with a soft toothbrush dipped in water. Remember to use a new ear swab or freshened toothbrush as the residue lifts away.
On purse hardware and girdles the green corrosion usually just lifts off. On vintage jewelry, it's often more stubborn, probably due to a different metal alloy.
For the fabric itself, I would look at rust removers or even a calcium-lime-rust type product. These may not be the right chemical "fighters" to work against the green build-up, but you'd have to know the metal's composition to determine what exactly would counter it. Whatever product you try, it would have to be diluted and applied sparingly, and rinsed well afterward, as rust removers are harsh.