In this paper we investigates the use of fractal geometry for segmenting speech signal. An overview of methods for computing the fractal dimension is presented focusing on an approach which makes use of the characteristic Power Spectral Density Function (PSDF) of a Random Scaling Fractal (RSF) Signal. FDS (Fractal Dimension Segmentation) is applied to a number of different speech signals and the results discussed for isolated words and the components from which these words are composed. In particular, it is shown that by pre-filtering speech signals with a low-pass filter of the form 1/k, they can be classified into fractal dimensions which conform to the correct range. This provides confidence in the approach to speech segmentation considered in this paper and in principle, allows a template matching scheme to be designed that is based exclusively on FDS.