It seems that $\Omega_{R[[x]]/R}^1$ is rather big. For example, take R to be the rational numbers. We can see that $\Omega_{R((x))/R}^1$ is a R((x))-vector space of infinite rank. As by results from Matsumura's book, a basis of $\Omega_{R((x))/R}^1$ coincides with a transcedence basis of R((x)) over R----remember that we take R to be the rationals which is of char 0. Yet this has nothing to do with the torsion in $\Omega_{R[[x]]/R}^1$. And what can we say for structures of $\Omega_{R[[x]]/R}^1$ for general R's?