Homotopy type of non-Cohen-Macaulay complexes

In a recent article Minimal Cohen-Macaulay Simplicial Complexes by Dao, Doolittle, and Lyle many examples of CM complexes (which happen to be minimal by their definition) are listed in Section 5. For example, included in the list are triangulations of $\mathbb{RP}^{2n}$ or the dunce hat. Some other interesting examples are A non-partitionable Cohen-Macaulay simplicial complex and A balanced non-partitionable Cohen-Macaulay complex which are recent counterexamples to the "partionability conjecture."

Edit: Since the question asks about both shellablitly and CM-ness it is perhaps worth noting that shellablity depends of the particular simplicial complex and not just the geometric realization. So, all spheres and balls are CM, but there are non-shellable examples of each.

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.

The order complex of a poset is the simplicial complex whose faces are the chains in the poset. So order complexes of combinatorially-defined posets are of the combinatorial flavor that you want, and the resulting complex is pure if and only if the poset is graded. But there are all kinds of examples of non-Cohen-Macaulay posets with natural definitions.

One typical kind of bad example is by subword or pattern containment. A particular instance of this is:

Sagan, Bruce E.; Vatter, Vincent, The Möbius function of a composition poset, J. Algebr. Comb. 24, No. 2, 117-136 (2006). ZBL1099.68081.

In this paper they look at all words of integers, ordered by subword containment. I guess that's graded by word length. Although the poset is infinite, intervals are finite, and typically are not Cohen-Macaulay.

Similar study has been made of permutation pattern containment by Peter McNamara, Jason Smith, Einar Steingrímsson and others. And here too, intervals often fail to be Cohen-Macaulay.