  In this paper, we are concerned with identifying among the family of posets
associated with Kohnert polynomials, those whose order complex has a certain
combinatorial property. In particular, for numerous families of Kohnert
polynomials, including key polynomials, we determine when the associated
Kohnert posets are (EL-)shellable. Interestingly, under certain diagram
restrictions, (EL-)shellability of a Kohnert poset is equivalent to multiplcity
freeness of the associated Kohnert polynomial.
