What guage is the super thin wire in headset cable (e.g smartphone)
This sort of wire is sometimes known as 'Magnet Wire' or similar names because it is frequently used for making inductors or motor coils. You can generally find it in many different wire gauges, usually coated with either enamel or polyurethane insulation.

This is one example that I have used before. It's 38AWG coated wire which is quite easy to remove the coating from with heat (you can also get it in different colours). There are many other options around.

You have to be careful though to check the current spec for this sort of wire. Given how small the diameter is, they are only usually rated for ~100mA or so at the most for the narrowest wire. Just make sure you're not going to be drawing more than its rated current to avoid heating (which could destroy the insulation if too extreme) or melting.