We deploy an adaptive observer recently developed for general hyperbolic partial differential equation (PDE) systems to detect and diagnose various drilling incidents. The well is modeled by a distributed PDE which, contrary to lumped models, preserves fundamental properties of well-flow dynamics enabling faster and more accurate incident detection and estimation. Wired drillpipe technology with pressure sensors is needed to locate and isolate incidents. Four realistic simulation case studies demonstrating various properties of the observer are presented. Drilling incidents treated in the simulation case studies include packoff in the annulus, formation in-flows, loss of circulation, and various combinations of these. Although simulation results show that the developed observers successfully estimate properties of the incidents that they are tailored for, they do not constitute an incident-detection system for drilling. However, they provide a part of the data on which an overall incident-detection system can rely.