Higher Specht polynomials were first defined by Ariki, Terasoma, and Yamada as a way to construct a basis of the coinvariant ring that respects the decomposition into irreducible S_n modules. They are higher degree versions of the ordinary Specht construction of the irreducible representations, obtained by applying a Young symmetrizer to an appropriate monomial. This construction has recently been generalized to form bases of several generalizations of the coinvariant ring. We will discuss the general theory of higher Specht polynomials in this talk, focusing on the question: When does the resulting polynomial vanish? Some surprises show up in the code, and we ask whether machine learning tools can help address this problem as well. Some of this material will be on joint work with Raymond Chou.