In my search for scholarly opinions on the subject, here is what I have found so far:
Al-Suyuti in his book al-Itqaan said that, 'twenty-one verses in the Qur'an were abrogated; some were agreed upon, while others are not. These abrogated verses are in the following Surahs: Al-Baqarah, Al-'Imran, An-Nisa', Al-Ma'idah, Al-Anfal, At-Taubah, An-Nur, Al-Ahzab, Al-Mujadilah, Al-Mumtahinah and Al-Muzzammil.
However, some scholars believe that no verses were abrogated.
There are scholars who are of the view that there are no abrogated verses in the Qur'an. A good, and authoritative, majority of the scholars, however, tend to say that such a viewpoint would be incorrect. These scholars uphold, as the most important point in their favour, the fact that the Qur'an itself has indicated this possibility. The relevant verses are 2:106 and 16:101
So, it seems that the majority (but not unanimous) opinion is that there are abrogated verses that survived in the text of the Quran. However, this passage claims that the main factor in the majority opinion is the existence of the two abrogation and substitution verses (2:106 and 16:101).
Dr. Mohammad Asad says that "there does not exist a single reliable Tradition to the effect that the Prophet ever declared a verse of the Qur'an to have been abrogated. At the root of the so-called 'doctrine of abrogation' may lie the inability of some of the early Commentators to reconcile one Quranic passage with another: a difficulty which was overcome by declaring that one of the verses in question had been 'abrogated'."
This opinion reiterates the information that there are no verses designated by the Prophet (PBUH) as abrogated, and it looks unfavorably at the majority conclusion about the existence of surviving abrogated verses.
The above referenced abrogation and substitution verses are: