While macroscopic properties of spin glasses have been thoroughly investigated, their manifestation in the corresponding microscopic configurations is much less understood. Cases where both descriptions have been provided, such as constraint satisfaction problems, are limited to their ground state properties. To identify the emerging microscopic structures with macroscopic phases at different temperatures, we study the p-spin model with p=3. We investigate the properties of self-sustained clusters, defined as variable sets where in-cluster-induced fields dominate over the fields induced by out-cluster spins, giving rise to stable configurations with respect to fluctuations. We compute the entropy of self-sustained clusters as a function of temperature and their sizes. In-cluster and out-cluster field properties support the observation of slow-evolving spins in spin models. These findings are corroborated by numerical studies in finite-size systems at low temperatures. Copyright © 2017 American Physical Society.