To be entirely frank, the popularity of this show saddens me. Inuyasha is certainly not terrible - it has a few good moments, the occasional flash of clever humour, and, unlike so many animes, dignity. However, it is utterly lacking in the essential elements of a worthwhile story. From the start, its premise dooms it to be stereotypical. The main plot centers around collecting the pieces of a shattered jewel before they can be possessed by evil, and is, as one would suspect, a totally generic epic fantasy affair. The story follows a familiar pattern of fighting off various enemies for pieces of the jewel, and is thus quite predictable, lacking in complexity, and easy to lose interest in. But as so many animes have shown, a poor premise can be rescued by deep, realistic characters. Sadly, no one rescues the story of Inuyasha. Kagome, the main character, is the stereotypical anime heroine (and far too reminiscent of Akane, the main character of the original comic author's previous work Ranma 1/2); she is kind to other females, but treats many males, especially her love interest, with unfair, unabashed, unjustifiable brutality. Inuyasha is a tough-on-the-outside-but-sweet-on-the-inside type, and Miroku is the lamentable stock character of "the pervert".
The flaws continue with what happens to this plot and these characters - namely, nothing. Despite constant action, the story does not progress. Despite regular romantic moments, neither does the main relationship. Despite ample time, the characters never really change. And to add a cherry to the sundae of mediocrity, all this stagnation is stretched into approximately 150 episodes.
My final criticism of this anime is the animation. While certainly not ugly, it displays almost disrespectful laziness on the part of the creators. The animators seem to take joy in long scenes of Inuyasha jumping through the air with wind whistling in which they have little to do but move a background.
In short, with all the beautiful animations of the world at one's keyboard-perched fingertips, there is absolutely no reason to watch Inuyasha.