We have studied the equilibria of liquid droplets wetting a step edge with an opening angle 0<α<π by a combination of analytical and numerical methods. Depending on the wetting properties of the substrate walls and on the liquid volume, different locally or globally stable liquid morphologies are found. Complete spreading of the liquid along the bottom edge of the step is observed at equilibrium contact angles θ satisfying θ<θs=π2−α2 . If the contact angle θ exceeds a threshold value θb the liquid exists in a blob-like configuration. Surprisingly, we find an intermediate regime at a sufficiently high liquid volume and in a range of contact angles θs<θ<θb , in which “cigar”-shaped configurations arise in addition to the blob. We close the paper by a detailed discussion of the stability of this novel liquid morphology.