Six ansatzes are investigated for their potential to allow three-dimensional (3D) ideal magnetohydrodynamic (MHD) equilibria. The ansatzes are based on a Clebsch representation for the magnetic field, B=∇H × ∇k, and a ‘generalized Clebsch representation’, B=∇×(∇K × ∇k), with ∇k being one of the coordinate directions of a cylindrical coordinate system. Three classes of equilibria, all with a straight magnetic axis, are obtained. Equilibria of the first class have a purely poloidal magnetic field of the Clebsch type with k=z and include the 3D equilibria already known. Equilibria of the other two classes have a purely toroidal (i.e. here longitudinal) magnetic field and pressure surfaces that can be chosen such that poloidal sections are closed. The second class is based on a Clebsch representation with k=θ. Solutions contain a free function of θ that determines the poloidal sections of the pressure surfaces at, say, z=0. The behaviour in the toroidal direction is then fixed but not periodic. For the third class, the generalized Clebsch representation with k=z is used. The equilibria are similar to those of the second class, with two important differences. They contain no free function and field lines are not planar. Finally, 3D vacuum fields, which exhibit 3D magnetic surfaces, are presented. They have the same geometry as the equilibria of the third class, and in fact can be obtained as a certain limit from these equilibria. Possible applications of the equilibria found are mentioned.