It is very difficult to prove or demonstrate reincarnation since it is not possible to replicate it in the lab, at least not yet. :)
Near Death Experiences (NDE) is another area which is still very controversial, yet it has been studied in scientific environments and numerous books have been written on this topic. Cases of NDE have been widely published in Mainstream newspapers. Below I'm providing links to some of the recent articles.
(Collection of the links below is credited to wanttoknow.info)
Child Psychiatrist Says Past-Life Memories Not So Uncommon in Kids
2006-07-25, ABC News
Kenneth Ring: 'You Never Recover Your Original Self'
1988-08-28, New York Times
Was a World War II pilot reincarnated in a body of a little boy?
2009-12-22, CNN Larry King Live
Heaven Is Real: A Doctor’s Experience With the Afterlife
2012-10-08, Daily Beast/Newsweek
Ask the Experts: What Is a Near-Death Experience?
2011-08-03, ABC News Nightline
Seeking Proof in Near-Death Claims
2010-10-25, Wall Street Journal
Doctor claims he has evidence of the afterlife
Parents Think Boy Is Reincarnated Pilot
2005-06-30, ABC News
Is There Such a Thing as Life After Death?
2010-01-22, Time magazine
Readers Join Doctor’s Journey to the Afterworld’s Gates
2012-11-26, New York Times
The Science of Heaven
2012-11-18, The Daily Beast/Newsweek
ABC: What it's like to experience a brush with death
2011-08-03, ABC15 (Phoenix ABC affiliate)
What Happens When We Die?
2008-09-18, Time magazine
The boy who lived before
2006-09-08, The Sun (highest circulation English language daily)
Washington Boy Says He Spoke to God After Flesh-Eating Bacteria Threatened Life
2011-08-02, ABC News Nightline
Near-death experiences: Heaven can wait
2009-03-31, The Independent (One of the UK's leading newspapers)
Ari Hallmark’s Story: To Heaven After the Storm
2012-05-07, WHNT-TV (Huntsville, Alabama CBS affiliate)
Bestselling Author and Pioneering Scholar of Near Death Experiences, Raymond Moody, Shares His Own Story in a New Memoir, Paranormal
2012-02-07, San Francisco Chronicle/PRWeb
Do loved ones bid farewell from beyond the grave?
Touching Heaven and Hell
2007-07-10, ABC News
How to give death a good name
2008-06-23, The Telegraph (One of the U.K.'s leading newspapers)
NDE demonstrates, that those who were declared clinically dead were still able to observe their environment, they could recall conversations, and describe rooms in which they were e.g. operated and had no prior knowledge of.
There are cases of children who can tell stories about certain historical events from the past and can recall names of people who were alive while those events took place. The early age of children from those cases and the little known historical events point to the fact that stories like these couldn't be created by those children from reading or hearing about them.
Here you can read about more NDE and past life stories.
We know that aura can be photographed. See video on The hisotory of aura imaging.
The next video is in Russian.
Doctor of technical sciences, Nikolai Zalicew says that not only you can take a picture of the aura, but also it is possible to weigh the soul. This was done through the experiment, which has been accomplished through the use of laboratory mice that were placed on super sensitive scale and then killed. Comparison of weight before and after the death of the mouse indicated that animal was slightly lighter. Energy signature of dying animal, was picked up by the gravitational field.
Similar experiments were done on humans, of course the measurements were done while the person was dying due to natural causes. The research in this area was led in 1907 by Dr. Duncan MacDougall of Massachusetts. It has been found that at the time of death, human body was lighter of about 3/4 ounces, which is equivalent to 21 grams.