I never wrote a review before in my life so please excuse my poor attempt, but after reading some of the reviews I just felt I had to give it a try.
I really thought this movie was very good. It's starts as the typical "missing wife, husband main suspect, no body yet found" tale we all have seen once in our lives. But after the first act it really gets extremely unsettling and special. I felt sick in between because of the horrible, frightening character of Amy. I can't describe it any further and think you should simply give it a try.
I now would like to address certain plot holes many people have been pointing out.
First of all, every movie has plot holes. I'm usually the first to notice, and even though there were some things that bothered me I got over them quite quickly since the movie had a tight grip on me.
Many are mentioning to different cameras in Desis house. First of all I know that many surveillance cameras overwrite their data after some time, so it is very likely that only the last few days (where she never left or entered the house) would be available to the police. Second of all, as far as I could see, the cameras only showed the outside of the house not what was happening IN the house. That's why she went near the window, which was within the perimeter of one of the cameras, to do her screaming attack. If you pay attention, you never see Amy appearing on any of the cameras when she checks them, only when she actually wants to be seen.
I could address many other so called "plot holes". I don't think they actually were plot holes per se, but made intentionally in order for us watchers to go..."But wait, how does she get away with this??". Think of the police officer who is in doubt of Amy's statements or even the lawyer.
What Fincher in my opinion wanted to demonstrate is the exemplary control Amy had over the public, were no questions were asked, nobody ever doubted her. It is kind of symbolic for witch hunts we ourselves have witnessed on TV or social media, were no second opinion is sought, nobody looks at the story from a second angle. Amy is the personification of the American Sweetheart, and is treated as such. It just shows us how blind and angry the crowd gets when you push certain buttons, Amy herself actually talks about that in the car scene. The pregnant intelligent wife, who was cheated by her husband. Or the poor girl next door who was always nice to the strange kid. Or the rich guy who thought he could have everything.
That is basically the story and that's why this film is so intriguing in my opinion. It's a mirror of how our society in the age of internet, social media, TV etc. functions. How we can be tricked into anger or sadness, and how little people actually question certain statements.