Derivation of Schroedinger’s Equation

There are various ways that the word “derivation” gets applied to a law or equation of physics. They all describe a way of getting to or motivating the result, but can range from just guessing on the basis of some analogy to rigorous mathematical proof from some other facts that have been previously accepted or which may seem more obviously “reasonable-seeming” or “fundamental”. I shall be collecting here and commenting on various approaches.