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 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 "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.