Finding a maximum independent set is a fundamental NP-hard problem. Given an unweighted graph, this problem asks for a maximum cardinality set of pairwise non-adjacent vertices. Some of the most successful algorithms for solving this problem use the branch-and-bound or branch-and-reduce paradigms. Results on branch-and-reduce algorithms were to a large part achieved by developing new, more practical reduction rules. However, other components that have been shown to have a significant impact on the performance of these algorithms have not received as much attention. One of these is the branching strategy, which determines what vertex is included or excluded in a potential solution. Even now, the most commonly used strategy selects vertices solely based on their degree and does not take into account other factors that contribute to the performance of the algorithm. In this work, we develop and evaluate several novel branching strategies for both branch-and-bound and branch-and-reduce algorithms. Our strategies are based on one of two approaches which are motivated by existing research. They either (1) aim to decompose the graph into two or more connected components which can then be solved independently, or (2) try to remove vertices that hinder the application of a reduction rule which can lead to smaller graphs. Our experimental evaluation on a large set of real-world instances indicates that our strategies are able to improve the performance of the state-of-the-art branch-and-reduce algorithm by Akiba and Iwata. To be more specific, we are able to compute a solution faster than the default branching strategy of selecting a vertex of highest degree on 90% of all instances tested. Furthermore, our decomposition-based strategies are able to achieve a speedup of 90% on sparse networks (33% on all instances).