Corrected entry: In episode two in the nightclub, after Visage have performed "Fade to Grey," their lead singer Steve Strange says, "I hope you enjoyed our new single, 'Fade to Grey.'" "Fade to Grey" was in fact a chart hit in the Autumn of 1980, so would have been old news in the Summer of 1981.
Corrected entry: In episode 2, Alex Drake and the property developer Moore go to the Blitz club in Covent Garden, London. This club is well documented as having a very strict dress code with reports of people being turned away for very obscure reasons. Even "making an effort" wouldn't have been enough to gain entry. Everyone in the club, except Alex Drake and Moore, was "dressed up" in what came to be called "New Romantic" outfits, including two (off duty) members of Alex's team. It would've been impossible for Alex Drake or Moore to have entered the club wearing the "normal" clothes they had been wearing in the previous scene in the wine bar.
Corrected entry: After Alex Drake is shot in the head, she seems to travel back in time to 1981 (that is the show's entire premise). One of the first clues Alex gets about being in 1981 is seeing a wall covered in posters advertising the single "Prince Charming" by Adam & The Ants. However, the rest of this episode is dated between 1st to 21st July 1981, and "Prince Charming" was not released until September 12th 1981.
Corrected entry: When Alex leaves the ladies - after chasing the mysterious clown figure - "Fade to Grey" by Visage comes to an end and Steve Strange (lead singer of Visage) can be heard in the background saying, "Hope you enjoyed our new single". The episode is set in the same week as the Royal Wedding (21st-29th July). However, "Fade to Grey" had charted in the top 40 @ No. 33 on the 17th January 1981 - six months earlier.