Sally Shepherd was 24 years old and the restaurant manager at the Young Vic theatre when she was murdered in December 1979.
She was brutally attacked and sexually assaulted as she walked home late at night after getting off a bus in Peckham, south London.
Her body was discovered in a builder's yard behind a police station.
Some 44 suspects were identified and eliminated from the original inquiry - among them the Yorkshire Ripper, Peter Sutcliffe.
However, police have now revealed they have a potentially vital clue - three strands of hair, which could well be from the killer.
The difficulty is the hairs are rootless.
Hairs without roots can be used to determine ethnicity, helping narrow down a field of suspects or confirm the identity of someone already in the frame.
But unlike hairs with a root, rootless hairs will not deliver a full DNA profile, in most cases.
"At the moment, that nuclear DNA that we normally find in the hair root is so degraded and so poor quality on a hair shaft that there is no way that we can normally get a result," says David Ballard, an expert in DNA analysis at King's College, London.
Without the guarantee of results, scientists are "very cautious" about examining hairs because there's a danger they'll use up the sample as it undergoes testing.
"Eventually you will run out of the evidence and this is something we do see in cases," says Dr Ballard.
It's certainly not a risk DI Stansfield is prepared to take, with only three strands of hair from the scene of Miss Shepherd's murder.
"They remain locked away until science is very much perfected on extracting the DNA from those," she says. "I can't sacrifice those."
She may have to wait three or four years, by which time scientists believe there'll be other key developments.
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