Together with Klaus Hulek we proved in 2011 that there is an effective algorithm which computes the Mordell-Weil group of X for ``most'' elliptic threefolds X with base P2. In the first part of the talk we explain what this statement means if one specializes to elliptic threefolds which are relevant for F-theory. Moreover, we explain several relations between singularity-theory invariants of the discriminant curve of an elliptic fibration and the Mordell-Weil rank of this fibration. In the second part we discuss extensions of these results to elliptic threefolds over arbitrary base surfaces and to certain classes of elliptic fourfolds.