Let $R$ be a regular local ring and $M$ a finitely generated reflexive $R$-module. When $R$ has dimension 2, then $M$ is a free $R$-module. This is discussed in Reflexive modules over a 2-dimensional regular local ring.
Suppose $R$ has dimension 3. Are there any results about the structure of $M$ in this case?
If $R$ is a Gorenstein (or even Cohen-Macaulay and Gorenstein in codimension $1$) local ring, then $M$ is reflexive if and only if it is a second syzygy (see e.g. this answer). As regular rings are Gorenstein, this directly extends the result you quote for regular local rings of dimension $2$. Indeed, a regular local ring of dimension $2$ has global dimension $2$, so second syzygies are free.
