We report on an extensive simulation study comparing eight statistical classification methods, focusing on problems where the number of observations is less than the number of variables. Using a wide range of artificial and real data, two types of classifiers were contrasted; methods that classify using all variables, and methods that first reduce the number of dimensions to two or three. The full feature space methods include linear, quadratic and regularized discriminant analysis, and the nearest neighbour method. The four dimensionality reducing classifiers are characterized by the transform they implement. The four transforms compared are the Fisher discriminant plane, the Fisher-Fukunaga-Koonz, the Fisher-radius, and the Fisher-variance transforms. The FisherFukunaga and the Fisher-radius transform based classifiers have recently been proposed for two class classification problems. We also present an extension to these transforms such that they can be applied to classification pro...