Operated by the Hungarian National Association of Radio Distress-Signalling and Infocommunications (RSOE), the Emergency and Disaster Information Service (EDIS) collects disaster information from around the globe, and disseminates it using the Common Alerting Protocol. They offer the data via web map, email, RSS, and Android application (an API is also available, though keys are apparently restricted to government organizations). There is a wide range of “disasters” included, but the normalized protocol supports filtering the events based on criteria such as scope, severity, and urgency. I use the Android application, configured to only show emergency-level alerts.
For earthquakes, I supplement RSOE EDIS with the USGS Earthquake Notification Service, which provides regional-based subscriptions. I subscribe to email alerts for earthquakes greater than 6.0 for all of the US, and greater than 4.0 for my local area.