Summary: The volume of a \(k\)-dimensional foliation \({\mathcal F}\) in a Riemannian manifold \(M^n\) is defined as the mass of the image of the Gauß map, which is the map from \(M\) to the Grassmann bundle of \(k\)-planes in the tangent bundle. Generalizing the construction by H. Gluck and W. Ziller [Comment. Math. Helv. 61, 177–192 (1986; Zbl 0605.53022)], ‘singular’ foliations by 3-spheres are constructed on round spheres \(S^{4n+3}\), as well as a singular foliation by 7-spheres on \(S^{15}\), which minimize volume within their respective relative homology classes. These singular examples, even though they are not homologous to the graph of a foliation, provide lower bounds for volumes of regular three-dimensional foliations of \(S^{4n+3}\) and regular seven-dimensional folations of \(S^{15}\), since the double of these currents will be homologous to twice the graph of any smooth foliation by 3-manifolds.