The electrohydrodynamic pattern forming instability (EHC) driven by an ac voltage applied to a homeotropically aligned nematic liquid crystal layer is experimentally studied near onset. By controlling an external magnetic field, many different scenarios become accessible. For finite fields various regular convective structures are observed, which we call, for example, wavy, chain, and bamboo-chevron patterns. The various types are documented with the help of a phase diagram governed by the frequency and the strength of the applied ac voltage. In addition the stability regimes in the voltage-wavenumber plane (the "Busse balloon") are mapped out. One finds significant differences from the conventional planar case, for which a theoretical analysis is lacking so far. For zero magnetic field, on the other hand, no regular structure is observed, even immediately above the onset of EHC. A new spatiotemporally chaotic pattern called the soft mode turbulence directly appears via a supercritical bifurcation from the nonconvective state. This is due to the presence of the Goldstone mode related to the spontaneously broken rotational symmetry associated with the director component in the plane of the layer.