We present a family of stochastic local search algorithms for finding a single stable extension in an abstract argumentation framework. These incomplete algorithms work on random labellings for arguments and iteratively select a random mislabeled argument and flip its label. We present a general version of this approach and an optimisation that allows for greedy selections of arguments. We conduct an empirical evaluation with benchmark graphs from the previous two ICCMA competitions and further random instances. Our results show that our approach is competitive in general and significantly outperforms previous direct approaches and reduction-based approaches for the Barabási-Albert graph model.