Surface electromyography (sEMG) is a non-invasive and straightforward way to allow the user to actively control the prosthesis. However, results reported by previous studies on using sEMG for hand and wrist movement classification vary by a large margin, due to several factors including but not limited to the number of classes and the acquisition protocol. The objective of this paper is to investigate the deep neural network approach on the classification of 41 hand and wrist movements based on the sEMG signal. The proposed models were trained and evaluated using the publicly available database from the Ninapro project, one of the largest public sEMG databases for advanced hand myoelectric prosthetics. Two datasets, DB5 with a low-cost 16 channels and 200 Hz sampling rate setup and DB7 with 12 channels and 2 kHz sampling rate setup, were used for this study. Our approach achieved an overall accuracy of 93.87 ± 1.49 and 91.69 ± 4.68% with a balanced accuracy of 84.00 ± 3.40 and 84.66 ± 4.78% for DB5 and DB7, respectively. We also observed a performance gain when considering only a subset of the movements, namely the six main hand movements based on six prehensile patterns from the Southampton Hand Assessment Procedure (SHAP), a clinically validated hand functional assessment protocol. Classification on only the SHAP movements in DB5 attained an overall accuracy of 98.82 ± 0.58% with a balanced accuracy of 94.48 ± 2.55%. With the same set of movements, our model also achieved an overall accuracy of 99.00% with a balanced accuracy of 91.27% on data from one of the amputee participants in DB7. These results suggest that with more data on the amputee subjects, our proposal could be a promising approach for controlling versatile prosthetic hands with a wide range of predefined hand and wrist movements.