Most people who have an interest in reincarnation have heard of the work of Ian Stevenson. He was a psychiatrist who latterly worked with children to investigate their claims to have lived before. He published a number of books about reincarnation and a number of books and television programmes have been written about his work. This is a lecture he gave in 2002.
His work has been criticised, mostly because it relies on anecdotal information, that he was reliant on interpreters who may have skewed the information they were relying, and because sceptics who doubt that reincarnation is a possibility because they do not believe in an eternal spirit, do presume that the content of the cases is bunkum.
If we start from the premise that reincarnation is impossible because the existence of an eternal soul is impossible (or not scientifically proven) it presents an insurmountable obstacle to any investigation, I think. You do not start an investigation into any phenomenon with a closed mind, because then you have drawn a conclusion without any evidence whatsoever, based on your understanding of what is possible, and not on the facts. It's not a starting point for investigation, it's an end to it.
Thus, I think any investigator has to begin with an acceptance that the phenomenon that they are setting out to investigate is a real phenomenon, which is all that Ian Stevenson did. When dealing with matters of belief, or memories, or human behaviour, ideas of absolute accuracy and scientific method have to be moderated with what it is possible to know from another human. It isn't possible to be exact in the way that science demands.
Stevenson's work considered certain markers that might strengthen a child's assertion that they were a reincarnation of another person. In some cases birthmarks eerily suggested the cause of death for the person that the child claimed to have been. In many ways it is that coincidence of both memory and physical evidence that seems the most compelling in the cases that he reported. His critics have said that he was gullible, taking evidence from other people without independently checking their assertions that a child had birthmarks, and ignoring the beliefs of the parents which might influence a child. Many of the families were from cultures which had a strong belief in reincarnation as a fact.
I was convinced by Stevenson's work in the light of my own experience. I cannot say whether it would have persuaded me if I had not had that experience of shaking hands with someone and suddenly receiving many impressions and memories which I later accepted were not from this life.
Recently there have been attempts to explain the physical processes which might lead to reincarnation as a part of evolution. I must admit that I have not made up my mind about the series of lectures by Todd Murphy which purport to explain reincarnation as a part of Darwinian evolution. Maybe this is a way forward for the research, to look at how it works... that at least may appeal more to the mechanical scientists who find spiritual research challenging when they don't believe in spirit.