A new edited collection of stories from the Nights has been published by Everyman (2014). It is a selection of stories collected by Wen-chin Ouyang. The stories are culled from Burton, Lane, Scott and Payne. Although I haven't received my copy yet it seems like an interesting text that would possibly make for a good intro reader. There seem to be hundreds of versions of the Nights coming out every week.
Here's an overview of the book's genesis by The National (Abu Dhabi) - http://www.thenational.ae/arts-culture/books/new-anthology-of-the-arabian-nights-has-controversial-subtext
from the article:
"“There is a large group of people who now see The Arabian Nights as a western rather than Arabic classic,” says the anthology’s editor, Wen-chin Ouyang. “And it is very possible that Aladdin and the Magic Lamp and Ali Baba and the Forty Thieves were European inventions from writers wanting to pen ‘Oriental tales’. They were then translated into Arabic and became part of this ever-growing compendium of stories by the 19th century. The story of Sindbad is really interesting, for example. Is it originally a Turkish folk tale, or a Turkish translation of an Oriental tale? There’s still work to be done on that.”