Studying the entanglement among many particles is a central topic of current research in quantum many-body systems and has great importance to the field of quantum metrology. In this context, the quantum Fisher information (QFI) provides a powerful framework for assessing, classifying and understanding the multipartite entanglement, and its metrological applications. Spinor Bose-Einstein condensates (BECs) constitute a particularly versatile platform for experimental investigations in this matter, however, a general and efficient experimental scheme for measuring the QFI in spinor BECs is still missing. The aim of this Master’s thesis is therefore to bring these two important concepts together by developing a protocol for the extraction of multipartite entanglement in spin-1 BECs. To this end, we derive an overarching framework relying on the computation of the QFI from quench dynamics and mean-field Bogoliubov theory. In a first study, we examine the strong influence of spontaneous symmetry breaking on the many-particle entanglement in single-mode BECs and compare our analysis to exact numerical calculations. We expand this study to an one-dimensional system by extending the Bogoliubov formalism to spin-1 quasi-condensates and deriving bounds for the entanglement of generic many-spin states. In this way, we compute the finite temperature amount of entanglement in a spin-1 BEC from dynamic susceptibilities after a weak quench. Our studies reveal the presence of multipartite entanglement in spin-1 BECs and the growth of correlations in regions close to quantum phase transitions. This work therefore paths the way towards the detection of multipartite entanglement in spinor BECs.