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After establishing the basic calculus and the construction and use of parametrices for elliptic operators, I suggest to go for Hörmander's theorem on propagation of singularities. This result is basic to understanding why high-frequency waves propagate along (geometro-optical) rays. Its proof (and, of course, its statement using wavefront sets) is a core example of microlocal analysis, and it shows that more useful things can be done with symbols than only taking their reciprocals. I find Hörmander's original proof in the ICM Nice 1970 proceedings very readable: Construct an operator which commutes with the given operator (the d'Alembertian, say) and (micro-)localizes to a neighbourhood of a given bicharacteristic, and then apply the $C^\infty$-wellposedness (established differently). Other proofs use microlocal energy estimates ("positive commutator method") or Egorov's Theorem (FIO calculus). Proofs of the theorem can be found, e.g., in the books by Grigis&Sjöstrand (CUP 1994), by Taylor (PUP 1981), by Unterberger (Aarhus Univ. 1976), and by Eskin (AMS 2011).