In a recent blog post, neurologist and skeptic Steven Novella explains the useful role played by anomalies in scientific discovery, but also why skepticism about the very existence of an anomaly is often an appropriate initial response.
Purveyors of pseudoscience often point to anomalies as evidence for their own pet hypotheses, which would typically both put an end to further inquiry and not result in further useful predictions. Genuine scientists look at anomalies as opportunities to question their assumptions--either assumptions embedded into their underlying theory, or assumptions about the equipment and research methods.
Novella's story about an alleged anomaly--namely, neutrinos which traveled faster than the speed of light--also handily illustrates the limitations of Karl Popper's falsificationist view of scientific methodology. Specifically, in the case of an apparent falsification of a scientific theory, there is always the possibility that the theory was not in fact falsified, but rather some auxiliary hypothesis that was used to set up the test. An attempt to test a theory also always involves background assumptions about one's equipment, methods, and so on. In the case of the neutrinos, the research team which reported the results was at first unable to discover any problem with their equipment or methods, but later they discovered a wiring problem that caused their measurement of the neutrinos' speed to be inaccurate.
I would love to be able to use this example in a class on the philosophy of science!
