But Elaine Doyle's killer John Docherty is now behind bars, put there by two pieces of evidence which officers did not even know they had collected.
With Greenock gripped in shock following the murder, it was quickly established that Ms Doyle had been to a disco in the Celtic Club in Laird Street, where she drank a single lager.
After leaving at 11pm, she visited a burger van before saying goodbye to her friends at midnight and setting off on the 13-minute walk home to Ardgowan Street.
Hours later, on the morning of June 2, 1986, her body was discovered in a lane, just 50 yards from her front door. She was naked, with the exception of a bra which was still attached to one arm. Her clothes were beside the body, but the bag she had with her was gone. Police believe the motive was sexual, but conclusive evidence that a sex attack took place was never discovered. There were clear marks on her neck indicating she had been strangled.
A blanket was taken from a police car and thrown over the body to protect it from public view, a blunder that would be seized upon in a trial nearly 30 years later as potentially contaminating the crime scene.
Members of the forensic team then used Sellotape-like strips in a bid to extract evidence from the 16-year-old's body, hoping they would pick up hairs or clothing threads that might offer a link to the culprit. When no such clues were found, the strips were sealed and all but forgotten about.
Police launched a high-profile appeal to trace Ms Doyle's clutch bag, believing it could be the key to unlocking the mystery. It turned up a week after a murder, on fire outside a public library with Ms Doyle's perfume used as an accelerant. The remains proved useless and no-one saw who left it. Despite extensive door-to-door inquiries and police questioning people with connections to Ms Doyle, there were few leads and no strong suspects.
As the years rolled by, policing entered a new age. Two years after Ms Doyle's death, the first murderer would be convicted with the help of DNA evidence. Within a decade the Scottish DNA database was set up.
The Elaine Doyle case retained its high profile and would occasionally be revisited in a bid to unearth new leads. A full forensic review of the case was ordered in 2005, by which time the use of DNA profiling in crime detection was widespread.
While the clothes found next to her body were in torn bags, meaning samples would be unreliable, the body tapings, sealed in acetate, were found to be in a perfect condition. Analysts discovered that crime scene investigators had also picked up DNA when they had examined Ms Doyle's body. A full profile was obtained from strips taken from her back, with the same man's DNA, mixed with the victim's own, found on samples taken from her face.
Believing they were finally close to solving the crime, police searched the database confident that if someone had carried out such a brutal killing, they would have committed other offences in the last 20 years. But the result came back negative - the suspect had never had his DNA taken.
A Crimewatch appeal, to mark 25 years since the murder, received a strong response but information from the public led nowhere. The breakthrough was to come as a result of trawling through old witness statements, with anyone named asked to give a voluntary DNA sample. Finally, after more than 700 people had been tested, detectives reached the name of John Docherty.
One man, who had passed away by then, had said in his witness statement that he had been at the Celtic Club with his friend, a John Docherty, before they both left at closing time. He had taken a taxi, the friend had said, leaving Docherty, who had never been spoken to in relation to the murder, to make his way home.
Docherty still lived in Greenock. When officers arrived at his home on May 12, 2012, he appeared an unlikely suspect. Living what appeared a normal life with his long-term partner and young daughter, he had joined the army in 1988, leaving six years later and taking a job with Inverclyde Council as a driver. Other than a housebreaking conviction as a teenager, his record was clean. When asked to give a DNA sample, he complied.
But two weeks later officers were told they had a hit - and that the chances of the result being wrong were one in one billion.
It was a huge development, but far from conclusive. A floor of Greenock Police Station, taken over by the case, was staffed by 36 officers and civilian workers including retired detectives brought in to offer expertise.
Docherty was seen as a strong suspect, but no more than that. His background was explored and it was found that a previous partner had experienced domestic abuse, which was never reported to police. But there could still have been an innocent explanation for how Docherty's DNA came to be found on Ms Doyle's body -perhaps they had danced together in the Celtic Club or were friends.
However, no witnesses picked Docherty's name from a list when asked who Ms Doyle had been seen with in the club, nor did her friends when asked who on the list had been known to the victim.
Martin Brown, who had passed Ms Doyle on the night she was murdered and always swore he could identify a man walking closely behind her, was contacted. He gave a positive identification of Docherty as the "angry looking man" he had seen in 1986.
The Crown Office agreed that Docherty should be detained for the murder of Ms Doyle. When he was finally brought in for questioning, interrogators, armed with a specially designed interview strategy, carefully probed him before hitting with the news that his DNA had been found on the victim's body. When asked how it could have got there, he could give no explanation and he was charged with murder. He did not express shock or distress.
After almost 4500 witness statements, 722 DNA samples, nearly 2500 house-to-house inquiries and 25 years, justice had caught up with Elaine Doyle's killer.