This paper introduces LABurst, a general technique for identifying key moments, or moments of high impact, in social media streams without the need for domain-specific information or seed keywords. We leverage machine learning to model temporal patterns around bursts in Twitter's unfiltered public sample stream and build a classifier to identify tokens experiencing these bursts. We show LABurst performs competitively with existing burst detection techniques while simultaneously providing insight into and detection of unanticipated moments. To demonstrate our approach's potential, we compare two baseline event-detection algorithms with our language-agnostic algorithm to detect key moments across three major sporting competitions: 2013 World Series, 2014 Super Bowl, and 2014 World Cup. Our results show LABurst outperforms a time series analysis baseline and is competitive with a domain-specific baseline even though we operate without any domain knowledge. We then go further by transferring LABurst's models learned in the sports domain to the task of identifying earthquakes in Japan and show our method detects large spikes in earthquake-related tokens within two minutes of the actual event.