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?