Today's social networks continuously generate massive streams of data, which provide a valuable starting point for the detection of rumours as soon as they start to propagate. However, rumour detection faces tight latency bounds, which cannot be met by contemporary algorithms, given the sheer volume of high-velocity streaming data emitted by social networks. Hence, in this paper, we argue for best-effort rumour detection that detects most rumours quickly rather than all rumours with a high delay. To this end, we combine techniques for efficient, graph-based matching of rumour patterns with effective load shedding that discards some of the input data while minimising the loss in accuracy. Experiments with large-scale real-world datasets illustrate the robustness of our approach in terms of runtime performance and detection accuracy under diverse streaming conditions.