Microwave Imaging (MI) has been extensively investigated for a number of years to develop a technique to detect breast cancer at the earliest stages of development. MI for this application is based on exploiting the dielectric contrast between normal breast tissue and cancerous tissue at microwave frequencies. Recent findings have reported overlapping dielectric properties between malignant and benign tumour tissue. Therefore, differentiating between benign and malignant tumours based on the dielectric properties alone is potentially problematic. Alternative methods for the classification of breast tumours have to be developed to improve the sensitivity and specificity of MI. In this paper, the Radar Target Signatures (RTS) of tumours are used to classify tumours, since the RTS contain valuable information about the shape and surface texture of tumours. The most important characteristics of any classification system are the feature extraction method used, along with the type of classifier itself. In this paper, the performance and robustness of three different feature extraction methods in combination with three classifiers are examined. A database of 352 tumours, comprising tumours of four shapes and sizes is created using Gaussian Random Spheres to evaluate the feature extraction methods. The three feature extraction methods - Principal Component Analysis (PCA), Independent Component Analysis (ICA), and Discrete Wavelet Transform (DWT) - are compared using three classification methods - Linear Discriminant Analysis (LDA), Quadratic Discriminant Analysis (QDA) and Support Vector Machines (SVM). Significantly, the performance of the feature extraction methods is evaluated using tumours in both a dielectrically homogeneous and heterogeneous breast models.