A coherent sheaf $\mathcal{F}$ over a Noetherian scheme $X$ is called (maximal) Cohen-Macaulay if $depth\_{\mathcal{O}\_x}(\mathcal{F}\_x) = \dim\mathcal{O}\_x$ for any $x\in X$, where $\mathcal{O}\_x$ is the local ring of $X$ at $x$.


Is there a simple example of $(X, \mathcal{F})$ such that $\mathcal{F}$ is Cohen-Macaulay but not locally free?


For regular schemes, I think they are equivalent. What about singular schemes? Under what conditions, a Cohen-Macaulay sheaf is locally free?



Cohen-Macaulay sheaves which are not locally free
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You are right. The Auslander-Buchsbaum-Serre theorem implies that the projective dimension of a CM module over a regular local ring is $0$ and hence a CM sheaf over a regular scheme is locally free.


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It is quite easy to give examples of non-locally free CM sheaves. 


**(a)**
It is relatively easy to prove that if $X$ is CM, then $\omega\_X$ is a CM sheaf.
So, take any $X$ that is CM, but not Gorenstein. Then $\omega\_X$ will be a non-locally free CM sheaf. Here is an explicit example:
$$X=\mathbb A^3/(x,y,z)\sim (-x,-y,-z)$$ 
See [this MO answer](https://mathoverflow.net/questions/55526/example-of-a-variety-with-k-x-mathbb-q-cartier-but-not-cartier/55529#55529) for a proof that $\omega\_X$ is not locally free. The fact that this $X$ is CM follows from that it is a finite quotient. 


**(b)**
Let $X$ be a normal surface (hence it is CM) and $\mathscr F$ an arbitrary reflexive sheaf of rank $1$. Reflexive sheaves are $S\_2$ and hence on a surface CM, but they're not always locally free. In fact, these sheaves correspond to Weil divisors while locally free sheaves of rank $1$ correspond to Cartier divisors. 


So for these sheaves there is a criterion you are looking for:
A reflexive sheaf of rank $1$ (which is CM on a normal surface) is locally free if and only if the associated Weil divisor is Cartier.


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I don't know if there is an elegant criterion for a CM sheaf to be locally free. There is one result that is sometimes useful:



> 
> Let $f:X\to Y$ be a morphism with equidimensional fibers. If $Y$ is regular and $X$ is CM, then $f$ is flat.
> 
> 
> 


In particular, if $f$ is finite, then $f\_\*\mathscr O\_X$ is locally free.


