In this talk, I will explain how one can extend (following Alexandroff and Tsikh) the notion of logarihmic forms and residues to that of logarithmic multi-residues along a complete intersection. I may sketch also the generalization to any Cohen Macaulay singularity. Then an analogue of the notion of freeness can be recovered through its caracterisation by maximal Cohen Macaulay property for the singular locus. I will then explain a result of Delphine Pol which Characterises the freeness by the fact that the module of logarithmic forms in degree equal to the codimension have minimal projective dimension.
In the last part of the talk I will show examples including the case of quasi homogeneous singularities of curves. I will also discuss potential applications to a notion of free objects in projective geometry, like arrangements of line or cuspidal curves in P^3, in the spirit of papers by Dimca or Faenzi-Valles.