There are several points of dispute when interpreting the Quran. For instance, 'some' clerics interpret the famous Nisa 34 verse of Quran as follows:
When a married couple go through problems in their relationship, they should talk to each other. If that doesn't work, separate your rooms. If this doesn't work either, the man can pound up the woman a little. If the problems persist, they should seek consult from outside - like the elderly whom would be listened to by both the man and the wife. If you still can't find any solution then get a divorce.
The translation of the word "darabe" constitutes the basis of the argument which say "The Quran lets men beat their wives" however there is so much lost in translation. The word 'darabe' has 17 different meanings one of which is a 'temporary separation'. This meaning, temporary separation, is also used in verse Nisa 101.
One should also know the context of which Nisa 34 was sent to Prophet Mohammed. It is said that a group of women have gathered around the prophet's house and they complained about their husbands' violence. In protest, they stayed at the masjid that night refusing to go back home to their husbands. In the morning the prophet calls for the husbands and comes down on them for beating their wives. Following this incident verse Nisa 34 was sent.
So, what does logic suggest when interpreting this verse? Beat your wives or separate houses for a while?
Nisa 34 actually reads as follows;
Step 1: Try talking to each other.
Step 2: Try separating your rooms for a while
Step 3: Try separating your houses for a while
Step 4: Seek outside counsel
Step 5: Get a divorce
Another point of dispute is around the hijab, the women's dress code. There are many verses in Quran that dictate Muslims of both genders to be modest in their dress and behavior. There are 3 verses in particular which directly talks about how women should cover themselves. These are particularly vague verses leaving a lot for interpretation and stirring a lot of dispute.
Samina Ali summarizes it quite well in er TED Talk.