The classical Hasse invariant is defined via the determinant of the Hasse-Witt matrix. It allows cutting out the so-called ordinary locus within the special fiber of a modular curve: this is the affine locus where the Hasse invariant is invertible. For more general Shimura varieties, the ordinary locus may be empty, and the Hasse invariant is then trivial. On the other hand, there exist for all Shimura varieties of PEL-type so-called generalized Hasse-Witt invariants which are vector-valued, but they are typically not robust enough to carry over the usual applications of the classical Hasse invariant. In this talk, we specialize to the scalar-valued cases that are most similar to the classical invariant (joint work with W. Goldring).