I know a lot of examples of classes of binomial ideals $I$ in $S=K[x_1,\dots,x_n]$ whose $S/I$ is a Cohen-Macaulay domain. Basically, if $I$ is a toric ideal and there exists a monomial order $<$ on $S$ such that the initial ideal of $I$ with respect to $<$ is squarefree then $S/I$ is Cohen-Macaulay.
So far, I have never met a class of non-toric ideals $I$ such that $S/I$ is a Cohen-Macaulay ring. Do you know some examples? Are they quite uncommon?
Thank you very much for your answers!