also see: Materials >> Conductive Threads
You can’t solder to most , but there are conductive threads that are actually made by spinning very thin flattened wire around a soft, flexible core, and these are great for soldering to.
In particular I did tests on a range of these threads from a company in Germany called Karl Grimm and these gave the best results. Their threads have Kevlar, and Vectran cores with very thin flattened wires wrapped around them. So the solder makes the connection to wire and not to thread, and that is why the results are so good. The threads are very soft and flexible, but the wires are visible and I imagine make them less fit for the sewing machine. Aesthetically they are very decorative.