As already discussed in class, the Analytic approach to Homer claimed to find many instances of "spurious" passages in the Iliad and the Odyssey.
One example in the Odyssey is the so-called "Continuation", i.e., the last part of Book 23 and all of Book 24. In Fitzgerald's translation, this is everything from p. 438, line 333 "opening glad arms to one another" to the end of his translation. A generally good treatment of the question is that given by D. S. Carne-Ross in his introduction to our edition of Fitzgerald's Odyssey, pp. lviii (last couple of lines on the page) - lxi. As Carne-Ross suggests, much of the "evidence" for the view that this whole section of more than 600 lines is spurious is that various ancient scholars referred to the reunion of Odysseus and Penelope as the τέλος (telos) of the Odyssey. The Greek word telos indeed means "end" - but it is often used in the sense of "end toward which something is directed, goal", and this may be what ancient scholars had in mind in their comments.
Another example of the same overall sort of argument involves a whole work, Prometheus Bound . We are not reading this play in the course - unfortunately, H&P merely give selections from it, which makes it hard to follow the intricacies of the plot properly. By and large, though, H&P's characterization (last sentence on p. 117, continued as first sentence of p. 120) of Prometheus Bound as displaying a negative, critical view of Zeus is certainly accurate. Particularly for this reason, as H&P state, pp. 117-120, some scholars have judged Prometheus Bound as "spurious", i.e., not by the author of the other six plays which we have by Aeschylus. Actually, though, as H&P point out, a single author might present different views, emphases, etc. at different times - and H&P therefore treat Prometheus Bound as being by Aeschylus. Also, another point - which H&P do not really consider - is that even in his other six plays, Aeschylus sometimes displays a more questioning, skeptical attitude toward Zeus and toward the gods in general than one might naively expect. (One such example is Agamemnon, lines 68-79, p. 563.)
Not infrequently, it is the conclusion of a work which modern scholars have somehow had problems with. Somehow, scholars feel they instinctively know how a work of art should end, and they consequently feel they can improve on what they find in our manuscripts. Even Carne-Ross, for example, in discussing (p. lxi in Fitzgerald's translation) the end of the Odyssey, seems to say that he could have written a better ending to the epic than Homer did - although ultimately Carne-Ross seems to bow to Fitzgerald's own assessment, pp. 506-507, that Odyssey, Book 24 is well crafted.
In addition to the end of the Odyssey, the final choral section in Oedipus Rex (lines 1458-1464 in H&P) has often been regarded as spurious. One of the supposed problems in this section of Oedipus Rex is the reference to "riddles" in the plural - supposedly, this is inconsistent with the overall plot of the story. For this point, though, cf. the material elsewhere in these study-materials concerning the Riddle of the Sphinx.
Another supposed problem is that the sentiment that no mortal can be judged happy until one can judge his entire life is so general as to be almost platitudinous in Sophocles' play. Exactly the same sentiment, for example, is clearly expressed in Herodotus' story of Croesus and Solon. The argument concerning Sophocles, Oedipus Rex is apparently that Sophocles himself would have been unlikely to have simply adapted Herodotus' point of view to his own presentation.