No, I believe that everything, after a long while, will return to its original state. Ashes to ashes, dust to dust. Humans were made out of the dust of the earth, and when we die, we return to that state. I do, however, believe that humans can leave things behind-memories, ideas, deeds, anything they did while they were living. Because we all affect the earth in some way, and by the end of our lives, it is somehow, if ever so slightly, effected by each individual's existence. These affects-memories, ideas that we have from a certain person- are often mistaken to be their actual spirit by those who knew the person while they were alive. So, if you were talking in a super-spiritual sense, you could say yes, spirits do exist because the ideas and memories we leave behind can never be killed. However, "Ghost Hunting" shows, I don't believe they're ghosts...they're either one of two things-really bad attempts at horror or actual cases of supernatural occurrences. The SUPERNATURAL is a whole different story from ghosts- it concerns beings that are beyond life, not natural, but still possessing an extreme presence. I believe in this: demons, which are essentially evil "spirits" that attach themselves to objects, locations, maybe even people (look up some of the most famous demonic possession cases, and you'll find some interesting stuff).
That's my opinion on the matter. I feel like people, a lot of the time, question "well why do you believe in 'demons' but not ghosts?" It's not a religious thing for me, its just more about logic and what I personally would define life to be. Just from experiencing it, I highly doubt its possible to fly around and haunt people when I die. However, the possibility that there exists concepts, forces and beings beyond life and nature doesn't seem that far-fetched to me. What reason do humans have to assume the world, the universe, the many universes and dimesnions, were created solely for them? That's like an ordinary person walking into an empty 30-story hotel and assuming every room is for them to sleep in, and not being surprised when the entire building is empty. To outright assume that concepts/beings don't exist beyond life is illogical.