Has it been scientifically proven that ghosts exist?
The short answer is that the Scientific community is on the fence when it comes to any Paranormal object, be it a ghost or otherwise. However, where once it was considered "fringe science", the serious investigation of Paranormal activity is now being pursued by many top Universities and Colleges, using scientific methodology and equipment. As we learn more, and more people with scientific backgrounds become involved, more theories on how ghosts manifest or where they exist will be proposed. Remember that Einstein's Theory of Relativity postulates an infinite number of alternate realities and many other things that most people don't believe in either - but are universally accepted by the scientific community, largely in part because many elements of his theory have been proven to be correct by scientific experiment.
Unknowledgeable people presenting bogus information as "evidence" has been a large problem for many people, as is just their own core beliefs, faith, or unwillingness to face the possibility that there's something other than Heaven or Hell. This was the problem in the past, and continues to be to this day. Groups claiming to be "Ghost Hunters" (unlike the TAPS group, which is a serious team) often put forth pictures and other "evidence" that is unclear and unverifiable at best. Few of these people have any real technical or scientific background, and each time it happens it creates even more skepticism. Even worse are the persistent hoaxers, and it's a lot easier today with moderate software to create effects that look good without closer inspection. I myself have analyzed purported EVP recordings which on the surface sounded real, but upon closer inspection were obviously faked. Same with pictures that are reported in the media; we looked at one case (myself and another Photoshop expert) where a picture was highly publicized at a local TV station. Upon close inspection, it was evident that the image was faked, albeit a very good one. Other things didn't add up as well in the entire story, and on the whole, it seemed more likely to be a ratings gimmick by the TV station as the reporter was present for the "photo".
Scientists, professional and amateur investigators, and just everyday people who have witnessed and seen ghosts are believers, but universal belief is based on evidence, and even with photographic, video and audio evidence, many people aren't yet willing to universally accept that ghosts exist. As one who has seen them, with evidence collected, I of course have no doubt, and it has caused me to rethink many things of a spiritual nature. There is a lot of evidence that tips the scale toward the existence of ghosts than there is to disprove they do. By evidence, I mean that which cannot be explained by any other cause.
I have no doubt that science will ultimately come to the realization of the existence of ghosts. Remember that science has often sworn that ancient species are extinct, only to be proven wrong when a specimen was found. The Coelacanth is a prime example of why no one should ever discount the possibility of something not being provable. Science long held the Coelacanth was extinct millions of years ago - only to be slammed when a live specimen was caught several decades ago.
We often consider ourselves smarter than we really are, and science, though often right, isn't always correct by a longshot. In the end, seeing is believing, and for anyone who has ever witnessed any type of entity, be it active or residual (a "recording" of an event that repeats and cannot interact with the viewer), there is no question of their existence. Remember too that even if science does prove their existence, not everyone will believe it either. There are still people who believe the Earth is flat, that we've never been to the Moon, and that the U.S. Navy supports an underwater alien base in the Bahamas. I can't tell you what the base is for in the Bahamas, but I've been there many times. I can say there isn't anything extraterrestrial going on, and like most highly classified military installations, activity there is kept secret for very good reasons, none of which has to do with aliens.
My advice is to do what many of us have done - start your own journey to determine for yourself if they exist or not, and keep an open mind when doing so. In the end, what YOU believe is really all that matters.