Magnet wire used to be available in a wide range of small sizes. The decline of hobbyist electronics has made fine copper magnet wire hard to find these days, though. Used to be Radio Shack and various mail order places had lots of fine sizes, down to about 34 or 36 gauge. Now, when I throw out any old piece of electronics I go through and look for coils, chokes and transformers with fine wire. This has been my source of magnet wire in fine sizes for about ten years now.
I do find Michaels craft stores to have an increasing selection of fine wire in their "beading" section. Not even sure what material these wires are except the obvious brass stuff.