Most interestingly defined (pure) simplicial complexes that occur in topological combinatorics are Cohen-Macualay. Some of these are even shellable (or have the homotopy type of a wedge of spheres; of same dimension). 
I would like to know examples of pure simplicial complexes (whose simplices are combinatorially defined) that are not shellable. 
Moreover, can anybody point out papers in which author(s) deal with simplcial complexes that do not have homotopy type of a wedge spheres?



Homotopy type of non-Cohen-Macaulay complexes
An interesting class of pure complexes which are not Cohen-Macaulay are the chessboard complexes. See <http://www.math.miami.edu/~wachs/papers/tmc.pdf>. Another interesting class of non-Cohen-Macaulay complexes are the $h$-complexes of Edelman and Reiner. See <https://arxiv.org/pdf/math/0311271.pdf>.


