We study the nonlinear stationary waves propagating in metal slot waveguides with a Kerr-type dielectric core. We develop two independent semi-analytical models to describe these waves in such waveguides. Using those models we compute the dispersion curves for the first ten modes of a nonlinear slot waveguide. For symmetric waveguides we find symmetric, antisymmetric, and asymmetric higher order modes which are grouped in two families. In addition, we study the influence of the slot width on the first symmetric and asymmetric modes. We also show that the dispersion curve of the first asymmetric mode is invariant with respect to the slot width for high propagation constant values and we provide analytical approximations of this curve. Using realistic values for the optogeometric parameters of the structure, we are able to obtain the bifurcation at a realistic power.