The formation of stable coalitions is a central concern in multiagent systems. A considerable stream of research defines stability via the absence of beneficial deviations by single agents. Such deviations require an agent to improve her utility by joining another coalition while possibly imposing further restrictions on the consent of the agents in the welcoming as well as the abandoned coalition. While most of the literature focuses on unanimous consent, we also study consent decided by majority vote, and introduce two new stability notions that can be seen as local variants of popularity. We investigate these notions in additively separable hedonic games by pinpointing boundaries to computational complexity depending on the type of consent and restrictions on the utility functions. The latter restrictions shed new light on well-studied classes of games based on the appreciation of friends or the aversion to enemies. Many of our positive results follow from the Deviation Lemma, a general combinatorial observation, which can be leveraged to prove the convergence of simple and natural single-agent dynamics under fairly general conditions.