In Episode 1 of Towers of Silence, DS Stephenson tells the 'Collector' that on February 2nd 1998 approx $2Million of Pentium 4 Processors were stolen. Intel didn't introduce the Pentium 4 until November 20th 2000.
In S1 Mel in the episode of a Simple Sacrifice says she is born in 1975, in S4 in the episode of Figue States when she talks about being adopted she is born in 1979 en at the end of S6 in the episode of Yahrzeit you can see on her gravestone that she is born in 1976.
In series 5, episode 1, the date on the CCTV camera shows "12 Sept 2005", but the CCTV shots from episode 2 show "15 Jan 2005", even though the plot leads us to believe the two episodes are chronological.
In episode Magdalene 26 S8 the both women claim they were 15 on January 14 1967 when both men died. However according to their birth register which can be seen by the nun who is writing this down in a book, the date of birth is may 1952, thus making the women 14 instead of 15 at the time.
In "Cold Fusion", the final episode of Series Five, Grace reads through Clifford Day's criminal record and mentions that, in 1979, he was charged with "conspiracy under the Data Protection Act". However, the UK did not get a Data Protection Act until 1984.
In "Towers of Silence" Boyd visits the freight terminal, apparently at London Heathrow Airport. In the background, a Ryanair plane can be seen on the tarmac - Ryanair do not fly into or out of London Heathrow Airport.
The goof item below may give away important plot points.
After the end of the investigation which results in Mel's death, Boyd draws a box with MEL in it on the glass notice board. However, this box can already be seen during at least one of the briefings held as part of this investigation.