I first came across the account of medium Estelle Roberts assisting police in the investigation of the disappearance of Mona Tinsley in 1937 England in the book Strange Talents by Bernhardt J. Hurwood. In that account, Ms. Roberts accurately described the murder, identified the killer, and located the body. A check of primary sources flavors those claims a bit differently.
Wikipedia has a good summary, and I've verified things like dates and places with contemporaneous newspaper accounts. Roberts involvement was covered at the time by Psychic News, and her direct account was published in her 1969 book 50 Years a Medium.
Mona Tinsley was reported missing in January of 1937, Roberts became involved a month later, and Mona's body was discovered the following June. Roberts did not ID the suspect, as Frederick Nodder was arrested a couple days after the disappearance, well before Roberts got involved.
Roberts also didn't help police locate the body, which was found by boaters. She says herself "“I can’t say precisely where" the body was, but supposedly channels Mona's tormented spirit to follow the path, which led to a field that Roberts and her entourage found impassible, though the killer supposedly had no problem following the route carrying the corpse.
The location she supposedly gave has along the lines of through fields and rivers, which describes pretty much all of England. It's implied that the body was found just beyond the "impassible" field - it was eight miles away across several villages, in an accessible area.
Roberts herself still hedged her bets, saying "The river holds the secret of the child’s whereabouts", which isn't the most exact way of saying you know the body is in the river.
So far, at best, even if Roberts had legit psychic powers, they didn't come in handy in this case, as the killer was already known and she didn't help locate the body. Most of the claims about the details of the murder were unverified, as Nodder never confessed.
Her knowing some limited details is less impressive when we learn that police questioned her about her impressions of the case prior to her entering the murder scene, which would tip her off. Her knowing the landmarks near a building she just travelled to isn't particularly remarkable. The biggest piece of evidence of her psychic powers was supposed to be the fact that she knew the killer used the side door to dispose of the body, which was proven because the front door was nailed shut. Despite having a 50/50 chance to begin with, she probably figured that piece out when she entered the house.
The only things that she reports to have known that were verified were that the child was dead and that she had been strangled. It was widely assumed Mona had been murdered - in fact this case led to changes in British law requiring a body for a murder charge. The strangling she was accurate on, but even then she only revealed this after the end of the trial.
Psychic News drip fed hints during the case, but only published details after they had been confirmed. According to the paper, Roberts told her followers that she had told the police where to find the body, which doesn't match her later account, and was told to her followers after the body had been found. Psychic News hints at other details, but doesn't publish any until after the trial was over.
Roberts reports being hesitant to assist police - contemporaneous accounts say she was concerned about the death penalty, and her later account says she didn't want publicity (despite announcing details to her followers at her own meeting hall). Either way, she claims that she helped in order to bring the family peace and closure.
Here again, even (or especially) if she was psychic, this doesn't hold up. Roberts claimed that Mona was raped (not verified in the autopsy) and that she was constantly in horror reliving the experience again and again, possibly for eternity. I'm sure this brought the family great comfort when they read this in the papers.
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