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As the failure of a hydraulic pump is always instantaneous, the failure data are difficult to obtain. High-efficiency fault diagnosis under small-sample conditions for hydraulic pumps is urgently required in engineering applications. A fault diagnosis approach based on wavelet packet transform (WPT), singular value decomposition (SVD), and support vector machine (SVM) is proposed in this study. First, the nonlinear, non-stationary vibration signal of the hydraulic pump is decomposed into components by WPT. Second, singular value vectors are acquired as feature vectors by applying SVD to the components. Third, the health states of the hydraulic pumps are determined and classified with a SVM classifier. Furthermore, the SVM and Elman neural network classifiers are compared in terms of fault classification to demonstrate the superiority of SVM in dealing with small-sample problems. The results of the plunger pump rig test show that the proposed method can diagnose the faults of the hydraulic pump accurately even when the number of samples is small.