An Old Fartz input here.
I use dies (and taps) as small as 3/0-120, which is 0.034 in. A fuzz smaller than 1 mm. Most times I use the die direct on wire. But, there are occasions where threading on the lathe is preferable. Where and when to do one or the other is mostly dependent on
you and what you are working on.
My lathe(s), both of them, will cut threads to 120 TPI. I, on the other hand, am not near that good. 64 TPI is as about as small as I will tackle. When I do cut threads with the lathe, I don't cut to full depth. Then finish with the die. The lathe cut threads are coarse but provide true concentricity to the wire. Once they are started, the die provides a smoother full depth cut.
Anything below 1/4 inch is cut manually. I rigged a hand wheel on the spindle, on the far outside of the head stock. 1/4-40 and smaller. Conceded, much(most) of my work below Nr 2-56 is in brass. But it works well enough for what I do, model building and instrumentation.
I don't recall the last time I had to cut fine threads in the larger sizes. It has been a while though,
many years . For that kind of work, it's vital to use the lathe. There are no taps and dies for the more specialty threads. But for small, more proportional threads it is well within the capabilities of a small machine.
My big lathe is a 12 inch machine, a Craftsman. Huge, but light weight. I use chucks adapted from a UniMat DB-200 for small work, below 3/16" / 5mm. Spindle thread M12X1. Even when I use a die with no guide cuts from the lathe, I still use the lathe to spin the work. And the die against a tail stock adapter. I don't have a tapping jig. Threading with a tap is done from the drill press, a tiny milling machine. With power off, obviously.
Sources: Taps and dies should be available from industrial suppliers down to 0-80 TPI. Smaller sizes from jeweler suppliers. I'm not too sure where, I haven't bought any for a long time. Fee Bay has metric taps and a "screw plate", a single plate with several threaded holes for making the male version. Sizes down to less than 1 mm. But I've only seen metric. USS sizes aren't listed. There was a supplier last year that had as small as 4/0. But I lost the address in a hard disk failure. But within 6 months.
Not a full discourse, but I hope I've added enough insight to be useful.
Bill Hudson