There are some blogs and threads on forums discussing inconsistencies available on the internet as it is.
Wikipedia covers some of them:
https://en.wikipedia.org/wiki/Disappearance_of_Madeleine_McCann"McCanns as arguidos
Early suspicion
Further information: § Media coverage
Praia da Luz, July 2009
The first indication that the media was turning against the McCanns was on 6 June 2007, when a German journalist asked them during a Berlin press conference whether they were involved in the disappearance.[117][118] On 30 June a 3,000-word article entitled "The Madeleine Case: A Pact of Silence" appeared in Sol, a Portuguese weekly, stating that the McCanns were suspects, highlighting alleged inconsistencies between their statements, and implying that the Tanner sighting had been invented.[119][120] The reporters had obtained the Tapas Sevens' mobile numbers and that of another witness, so it was apparent that the inquiry had a leak.[45][118][121]
This and later articles in the Portuguese press, invariably followed up in the UK, made several allegations, based on no evidence, that would engulf the McCanns for years on social media. They included that the McCanns and Tapas Seven were "swingers", that the McCanns had been sedating their children, and that the group had formed a "pact of silence" regarding what had happened the night of the disappearance.[100] Much was made of apparent inconsistencies within and between the McCanns' and Tapas Seven's statements. The police had asked the group questions in Portuguese, and an interpreter had translated the replies. The statements were then typed up in Portuguese, and verbally translated back into English for the interviewees to sign. Misunderstandings were inevitable.[62][100]
Among the inconsistencies was whether the McCanns had entered the apartment by the front or back door when checking on the children. According to the Polícia Judiciária case file, Gerry stated during his first interview, on 4 May 2007, that the couple had entered 5A through the locked front door for his 21:05 and her 22:00 checks, and in a second interview, on 10 May, that he had entered through the unlocked patio doors at the back.[122] (The patio doors could only be unlocked from inside, so the parents had left them unlocked to let themselves in.)[43] There was also an inconsistency about whether the front door had been locked.[122] Gerry told the Sunday Times in December 2007 that they had used the front door earlier in the week, but it was next to the children's bedroom, so they had started using the patio doors instead.[46]
The Polícia Judiciária questioned why, when Kate discovered Madeleine was missing, she had run to the tapas restaurant leaving the twins alone in 5A, when she could have used her cell phone or shouted to the group from 5A's rear balcony.[123]
Bedroom shutter
Officer dusts children's bedroom shutter
without protective clothing.[46]
—The Daily Telegraph
Another issue was whether the exterior shutter over Madeleine's bedroom window could be opened from outside. According to journalist Danny Collins, the shutter was made of non-ferrous metal slats on a roller blind that was housed in a box at the top of the inside window, controlled by pulling on a strap. Once rolled down, the slats locked in place outside the window and could only be raised using the strap on the inside.[124] Kate said the shutter and window were closed when Madeleine was put to bed, but open when she discovered Madeleine was missing. Gerry told the Polícia Judiciária that, when he was first alerted to the disappearance, he had lowered the shutter, then had gone outside and discovered that it could be raised from the outside.[125][126] Against this, the police said the shutter could not be raised from the outside without being forced, but there was no sign of forced entry; they also said forcing the shutter open would have caused a lot of noise.[124]
The apparent discrepancies contributed to the view of the Polícia Judiciária that there had been no abduction.[127][128] Even Kate's shout of "they've taken her" was viewed with suspicion, as though she had been paving the way for an abduction story.[46] Particularly from August onwards, these suspicions developed into the theory that Madeleine had died in apartment 5A as a result of an accident—perhaps after being sedated to help her stay asleep—and that her parents had hidden her body for a month, before retrieving her and driving her to an unknown place in a car they had hired over three weeks after the disappearance.[4][129] In 2010 Carlos Anjos, former head of the Police Detectives Union in Portugal, told BBC Panorama that most Portuguese investigators still believed Madeleine died as a result of an accident in the apartment.[130]"