The BBC generously provided TV crime addicts a choice of destination on Sunday night. While BBC One’s Death in Paradise wafted viewers off to a palm-fringed tropical paradise, BBC Two’s Forensics: The Real CSI took us to a waste-disposal site in Coventry.

This is the fourth series of the sober, unobtrusively made but invariably gripping documentary about the sort of crimes that are too sordid for cosy crime dramas but too inconclusive for anything a bit more sinister (there’s little worse than watching a murder mystery for six weeks only for there to be no murderer).

In this opening episode, a badly decomposed body had been discovered in a freezer dumped at the skip yard. West Midlands Police were faced with a dual mystery: who was the man in the freezer – and who put him there?

The first question required the forensics bods, kitted out in white overalls and firing away with their cameras just like Emilia Fox in Silent Witness. In fact, the programme would benefit any crime fiction writer in search of realistic details, as the team discussed the importance of maggots in ascertaining the time of death, the precise discoloration of the skin after death (green to darker green and then black) and so forth.

Meanwhile, tenancy records provided detectives with a name: 71-year-old John Wainwright. A visit to his bleak, now emptied flat in a Birmingham tower block revealed the cupboard that had housed the freezer, this evidently having leaked its contents after the electricity had been cut off. The neighbours said they had twice complained about the smell, only to be ignored by the building’s administrators. Modern Britain in a nutshell.

While the forensic pathologist examined the dead man’s organs to see whether he might have expired from natural causes, an odontologist used a 3D scan to match a Facebook snap of John Wainwright with the dead man’s skull and teeth. As a memento mori, this superimposing of a skull on to a smiling, living face was second to none.

The victim apparently had no friends or family, and without knowing anything else about him, I couldn’t help feeling that Wainwright was now receiving more attention in death than he had in life. But it was later discovered that he had a live-in carer, 50-year-old Damion Johnson, who though unrelated, referred to Wainwright as “dad”. When questioned by police, it was revealed that it was Johnson who had put Wainwright in the freezer after his death, claiming that he was so overcome by grief that he “didn’t want to let him go.”

Rather than a spontaneous act, however, Johnson had purchased the freezer at the time of Wainwright’s death. He used Wainwright’s credit card to steal £17,000 and by not reporting his demise, he continued to claim a carer’s allowance and to live in his flat.

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The police now had a motive but they had to try and ascertain whether Johnson murdered Wainwright (a broken cheekbone and teeth suggested that he might have) or was simply guilty of fraud and preventing a lawful burial. With insufficient evidence, the Crown Prosecution Service eventually decided to go with the lesser charges and Johnson was given a two-year prison sentence.

As with the previous three series of Forensics: The Real CSI, this was coolly (almost forensically, you could say) narrated by Siobhan Finneran – and the Open University co-production was generally a model in un-exploitative true-crime filmmaking.

The professionals also came across as both matter-of-fact and impressively dedicated. “It matters to do the right thing when someone dies,” said the forensics coordinator, Sonia Parkinson. “That’s why I do this job.”