A one-component surface wave has a displacement polarization vector in a single direction. Alternatively, it can be thought of as originating from a single evanescent bulk wave, and therefore cannot exist in isotropic solids. It has recently been established that the existence of one-component surface waves is compatible with material symmetry and that they can occur even in transversely isotropic materials. This paper deals primarily with the existence of one-component surface waves in materials with a plane of symmetry normal to the direction of propagation, with particular attention given to transversely isotropic materials in this class. A unique feature of these waves is that the traction vector which vanishes at the free surface is zero at all depths, implying that one-component surface waves are also pure modes of slabs. All of the "theoretical materials" which support the wave have the unusual property that at least one Poisson's ratio must be negative.