R' Kasher in Torah Shelemah - addenda to Parshas Shemos -attempts to prove that the Ibn Ezra to Sefer Shemos ((Pirush HaAruch)-the long recension)) was written by his students[1a]. He cites a proof the fact that this commentary cites information from Karaim  that already have a source in Chazal and that it is unlikely that the IE (who was no friend to the Karaites) should have done this. See there at length.
In the verse that discusses the prohibition of eating חלב (certain fats) the Ibn Ezra records an extensive debate that he had with a Karaite. Basically he forces the Karaite to agree that Anan's maxim חיפוש במקרא הרבה is simply insufficient and that in the end even the Karaites must accept the Kabbalah of the Rabbis.
It seems logical to suggest that the Ibn Ezra purposely chose to cite from Karaite scholars explanations that originate in Chazal. This way he could emphasize that even the Karaite Gedolim essentially agree with the interpretations of Rabbinates.
In an article in the Sefer Zikaron L' Zecher R' Yitzchok Nissim, R' Nachum Eliezer Rabinowitz (better known for his commentary on the Rambam the Yad Peshuta) discusses one of the more perplexing issues that students of the Yad come across. The Rambam consistently cites reasons for halachos that are different from those given by the Talmud (e.g. Muktzeh,Kiddush HaChodesh,etc.). Rabinowitz's explanation is that the Rambam left out the reasons of the Talmud because the scholar is assumed to know them already and rather chose to add his own explanations to resolve other unseen difficulties in the text See there at length.
I find this interpretation difficult. First, the Rambam writes in his letter to R' Pinchas Dayan that he never added in anything from his own pilpul unless it is designated by a יראה לי. What Rabinowitz describes certainly sounds like Pilpul to me. Second, the Rambam writes in his Introduction to the Yad that in writing the Yad he intended to write a complete digest - so one should not have to pick up any sfer besides for this one . He certainly could not have left out large amounts of information as Rabinowitz posits. Third, in most cases the Rambam does supply the standard Talmudic reason.
R' Zevin in his review of R' Kasher's "Ha'Rambam V'haMechilta D' Rashbi"  mentions that a "popular view that the Rambam ignores Chazal's exegesis and creates his own instead" has now been disproven. This is because now it is clear that the Rambam used this lost Mechilta as a source for many Halachos (Kasher counts more then one hundred). This being the case we cannot rule out the possibility that the Rambam had access to other Midrashei Chazal that we might not know of.
Particularly interesting is R' Chaim Tchernowitz (Rav Tsair)'s explanation in Toledot HaPosekim . He refers us to the aforementioned letter to Pinchas Dayan in which the Rambam writes that "he purposely chose to leave the names of the Tannoim out of the Yad in order to refute the Karaites - because they say that your Kabbalah is from only a few Talmidei Chachomim (Yechidim)..."  Noting a definite Anti-Karaite intent in the Rambam, Rav Tsair proceeds to demonstrate that in many of the Halachot in which the Rambam expands on Chazal's explanation or provides his own explanation - there is a parallel polemical passage against these Halcahot in the Karaite works. This is not a panacea and does not answer all the problematic passages but it remains a fascinating starting point.
See also the post "On the Main Line" on Karaism I and II.
The Radbaz has several interesting Teshuvos on Karaim in one of which he describes his conversations with the leaders of the Bnei Mikra.
[1a] I recall that one of the editions of Tzefunot contained an article arguing for this point based on a signature in one of the manuscripts.
 For example, he cites the Karaite Yefet Ben Ali forty-two times in his commentary of the minor prophets. See Wikipedia- Karaites "Though he opposed Karaism, the Rabbinic commentator Abraham Ibn Ezra regularly quoted Karaite commentators, particularly Yefet ben Ali, to the degree that a legend exists among some Karaites that Ibn Ezra was ben Ali's student."
 But see the letter to R' Pinchas Dayan mentioned previously and also the Maharitz Chajes's heated defense in Tifferet L' Moshe (his polemic against Shadal - The Yeshuos Yaakov's Haskamah is also worth reading.).
 Interestingly, R' Avraham Ben HaGra had already divined the Mechilta's existence in Rav Peolim. See also R' Mordechai Gifters review in the YU journal Talpiot and R' Kasher's response.
 Rav Tsair's knowledge of Karaite seforim stands him in good stead. He also notes Anti-Karaite polemics in the Sheiltos D' R' Achai and Halachot Gedolos as well as other seforim.
 See also the Chakira article on Dealing with Religious Dissenters.