what inequalities can one have between $\operatorname{depth} R$ and $\operatorname{depth} M$? Let $(R,m)$ be a commutative Noetherian local ring which  is not CM. Let $M$ be a finite $R$-module.   

what inequalities can one have between $\operatorname{depth} R$ and $\operatorname{depth} M$?    

Obviously there are many examples for the case $\operatorname{depth} R \gt  \operatorname{depth} M$.   

Can you give an example of a finite $R$-module, $M$, in which $\operatorname{depth} R \lt \operatorname{depth} M$?  

 A: Take $R := {\mathbb k}[[x,y]] / (xy,y^2)$ and $M := R / (y)$. Then $\text{depth}_R(M) = 1$ while $\text{depth}_R(R)=0$.
For modules $M$ of finite projective dimension over $R$, you have:

Auslander-Buchsbaum formula: If $\text{pdim}_R(M)<\infty$, then $$\text{depth}_R(R) = \text{depth}_R(M) + \text{pdim}_R(M).$$

In particular, in this case we always have $\text{depth}_R(M)\leq\text{depth}_R(R)$. 
The Auslander-Buchsbaum formula continues to hold when $\text{pdim}_R(M)$ is replaced by $\text{gpdim}_R(M)$, the Gorenstein-projective dimension. You have $\text{gpdim}_R(M)=\text{pdim}_R(M)$ in case $\text{pdim}_R(M)<\infty$, but $\text{gpdim}_R(M)$ can be finite even when $\text{pdim}_R(M)$ is not: For example, $\text{gpdim}_R(M)<\infty$ for any finite $R$-module $M$ if (and only if) $R$ is Gorenstein.
The example from the beginning of the answer shows, however, that there cannot be an "ultimate" notion of (non-negative) dimension for which the analogue of the Auslander-Buchsbaum formula is true for all modules.
