I read recently a comment by a philosopher that Karl Popper's "falsifiability" theory is considered obsolete. Is this so? I always found it to be quite useful. If it's obsolete, what rendered it so, and by what was it replaced?

I'll add a third problem to Popper's views... it classifies obvious psuedo-sciences as sciences such as astrology, so long as they make potentially falsifiable predictions. Furthermore, it does nothing to distinguish something radically implausible like astrology from something more plausible, but not falsifiable such as ad-hoc psychological analysis. Popper's views and others similar to it (verificationism and logical positivism) belong to an era of philosophy when it was believed philosophy could be made 'scientific'. It has not really been replaced because few philosophers still hold to that belief.