As a prospective key technology for the next-generation wireless communications, reconfigurable intelligent surfaces (RISs) have gained tremendous research interest in both the academia and industry in recent years. Only limited knowledge, however, has been obtained about the channel eigenvalue characteristics and spatial degrees of freedom (DoF) of systems containing RISs, especially when mutual coupling (MC) is present between the array elements. In this paper, we focus on the small-scale spatial correlation and eigenvalue properties excluding and including MC effects, for RISs with a quasi-continuous aperture (i.e., holographic RISs). Specifically, asymptotic behaviors of far-field and near-field eigenvalues of the spatial correlation matrix of holographic RISs without MC are first investigated, where the counter-intuitive observation of a lower DoF with more elements is explained by leveraging the power spectrum of the spatial correlation function. Second, a novel metric is proposed to quantify the inter-element correlation or coupling strength in RISs and ordinary antenna arrays. Furthermore, in-depth analysis is performed regarding the MC effects on array gain, effective spatial correlation, and eigenvalue architectures for a variety of element intervals when a holographic RIS works in the radiation and reception mode, respectively. The analysis and numerical results demonstrate that a considerable amount of the eigenvalues of the spatial correlation matrix correspond to evanescent waves that are promising for near-field communication and sensing. More importantly, holographic RISs can potentially reach an array gain conspicuously larger than conventional arrays by exploiting MC, and MC has discrepant impacts on the effective spatial correlation and eigenvalue structures at the transmitter and receiver.