We study steady solutions of the two-dimensional inviscid Boussinesq system in a periodic channel. Our main interest is to understand when steady states near shear flows are rigid, and when nontrivial nearby structures can occur. We present several rigidity and flexibility results, highlighting the roles of stable stratification and stagnation points. In particular, we discuss situations in which laminar steady states are forced to be shear flows, as well as regimes where non-shear and even non-laminar steady states can bifurcate from shear profiles. This is a joint work with Marc Nualart (ICMAT) and Javier Penafiel-Tomas (EPFL).