Let $X\subseteq Y$ be a (singular) complex analytic subspace or closed subscheme with defining ideal $I$, where $Y$ is smooth. Stalks of the structure sheaf of $X$ are assumed to be integral noetherian domains. The conormal sheaf $I/I^2$ of $X$ in $Y$ is locally free if and only if $X$ is a local complete intersection.
I am wondering if there are well known conditions on $X$ such that the double dual $(I/I^2)^{**}$, or equivalently, when the normal sheaf $(I/I^2)^*$ is locally free. The subspace $X$ may also be assume to be normal, if necessary.
I guess that it should be a less restrictive assumption than local complete intersection if only the dual of the conormal sheaf is locally free -as the dual is already a reflexive sheaf. But I have not be able to find a nice condition that ensures this.