In neuroscience, synchronization processes are increasingly recognized as key mechanisms in different aspects of global brain functions and brain disorders. Research into these large-scale processes has been fueled by brain imaging data and the availability of the structure of the human connectome. We investigate the synchronization of phase oscillators located at the network nodes (brain regions) and interacting along the weighted edges (neuronal bundles connecting them) in the human connectome core network. It comprises the central part of the connectome associated with the eight hubs---key nodes that transmit information between different brain parts. We consider both the network with its natural (empirical) edge weights, which affect interactions among oscillators, and its binary version dominated by topology features. We show the results of the topology and spectral analysis of these networks and simulations of the synchronization of Kuramoto oscillators when the strength of the global pairwise coupling varies from significant negative to large positive values. The emergence of partial synchronization as a relevant pattern in human connectomes has been observed at negative pairwise couplings, causing the phase frustration effects, and at positive couplings below the master-stability threshold, where the cluster synchronization mechanisms become active. Identifying nodes in these clusters by the Laplacian eigenvector localization methods reveals the role of brain hubs in these processes. Both partial synchrony patterns induce multiscale oscillations in the global phase order parameter, which we study using multifractal analysis. We show how the related singularity spectra depend on the coupling strength and sign and the weights of the edges.