Higher-order tensor decompositions have hardly been used in muscle activity analysis despite multichannel electromyography (EMG) datasets naturally occurring as multi-way structures. Here, we seek to demonstrate and discuss the potential of tensor decompositions as a framework to estimate muscle synergies from $3^{rd}$-order EMG tensors built by stacking repetitions of multi-channel EMG for several tasks. We compare the two most widespread tensor decomposition models -- Parallel Factor Analysis (PARAFAC) and Tucker -- in muscle synergy analysis of the wrist's three main Degree of Freedoms (DoFs) using the public first Ninapro database. Furthermore, we proposed a constrained Tucker decomposition (consTD) method for efficient synergy extraction building on the power of tensor decompositions. This method is proposed as a direct novel approach for shared and task-specific synergy estimation from two biomechanically related tasks. Our approach is compared with the current standard approach of repetitively applying non-negative matrix factorisation (NMF) to a series of movements. The results show that the consTD method is suitable for synergy extraction compared to PARAFAC and Tucker. Moreover, exploiting the multi-way structure of muscle activity, the proposed methods successfully identified shared and task-specific synergies for all three DoFs tensors. These were found to be robust to disarrangement with regard to task-repetition information, unlike the commonly used NMF. In summary, we demonstrate how to use tensors to characterise muscle activity and develop a new consTD method for muscle synergy extraction that could be used for shared and task-specific synergies identification. We expect that this study will pave the way for the development of novel muscle activity analysis methods based on higher-order techniques.