Has a new functional structure ever been observed arising spontaneously?
It seems like you are attempting to frame this as a question about current evolutionary theory. However, current evolutionary theory doesn't expect anything as complex as an organ to arise spontaneously without any precursors. You suggest a new claw in an insect would be something "spontaneous" enough for you. This again shows that your question and the answer you are looking for are not related to evolutionary theory. The insect organs you are talking about are all modified appendages. This includes claws, mandibles, antennae, legs, fangs. These appendages all share a common genetic and developmental framework that has been slightly modified over long periods of time to give raise to the variety of insect appendages that we see today. 

The idea that evolutionary theory supports the spontaneous creation of organs is a willful ignorance of evolutionary theory. The molecular mechanisms that generate novelty (e.g. genome duplication, gene duplication, de novo genes, mutation, gene regulation) for natural selection to work on are relatively well understood. Since we have an understanding of how novelty is created it is known that spontaneous organ generation has never happened or is expected to happen under evolutionary theory. 

Here is link to the University of California, Berkeley's webpage designed to teach the public about evolution.