Let \((R,\mathfrak{m})\) denote a local ring that is a factor ring of a local Gorenstein ring. Let \(M\) denote a finitely generated \(R\)-module. Let \(K(M)\) denote the canonical module of \(M\) (see the reviewer in [J. Algebra 275, No. 2, 751--770 (2004; Zbl 1103.13014)]). In the reviewer's paper [loc. cit.] \(M\) is called canonical Cohen-Macaulay module whenever \(K(M)\) is a Cohen-Macaulay module (CCM). Note that \(M\) is CCM if \(M\) is Cohen-Macaulay while the converse does not hold. In the present paper the authors prove a lifting result saying that \(M\) is CCM whenever \(M/xM\) is CCM if \(\dim M \geq 4\) and \(x \in \mathfrak{m}\) is a strict \(f\)-element with respect to \(M\), i.e. \(x\) is filter regular with respect to all non-zero modules of deficiency of \(M\). In the second part they study the property of being CCM in terms of the polynomial type introduced by \textit{Nguyen Tu Cuong} [Nagoya Math. J. 125, 105--114 (1992; Zbl 0783.13020)]. Moreover they describe the non-CCM locus \(\text{nCCM} = \{\mathfrak{p} \in \text{Spec} R | M_{\mathfrak{p} } \text{ is not CCM } \}.\) If \(\dim M \leq 4\) it is closed. For \(d \geq 5\) there are reduced geometric local rings \(R\) such that \(\text{nCCM}(R)\) is not stable under specialization.