Singer (made in USA) should be an SAE thread. Could be fine or coarse thread pitch. My local ACE hardware has a tremendous selection of screws, bolts, and such. Lowe's stocks stainless steel machine screws in size 4-40, 6-32, and 8-32. A hobby shop that caters to radio control enthusiasts will have small screws in both SAE and metric threads.
Grainger, Fastenall, and McMaster-Carr also stock a wide variety of hardware. If you can identify the size and thread pitch, a gunsmith may be able to help.