According to Wikipedia, and common understanding1, the definition of a ghost is the following:
In traditional belief, a ghost is the soul or spirit of a deceased person or animal that can appear, in visible form or other manifestation, to the living.
As such, before we can scientifically investigate such a phenomenon, we need to understand what the soul is. Without a working definition of soul, it is not clear how we could discern natural manifestations from real ghosts.
Now, the soul is the transcendant part of a human being, in other words it can be described as what is left of a person if you take away the body.
Soul, in religion and philosophy, the immaterial aspect or essence of a human being, that which confers individuality and humanity, often considered to be synonymous with the mind or the self.
Practically speaking, there has to be a soul for ghosts to exist at all, and the soul must be separable from the body, through death.
Belief in ghosts is based on the ancient notion that a human spirit is separable from the body and may maintain its existence after the body’s death.
Now - what problems do the previous definitions pose to scientific research?
First of all, any working definition of the soul is strictly unscientific: the soul is intended to be non-corporeal (or in other words, pretty much unmeasurable). So we are left with little of no means of substantiating a claim on ghosts - they can only be believed in, but not studied via physical detection.
Secondly, as any soft, non-scientific concept, it is nearly impossible to define: on the wikipedia page there are no less than 22 different religious definitions of the soul.
Lastly, all our scientific investigation on the workings of the mind through neuroscience has been quite effective in discovering many examples of parts of our consciousness which are physical in origin and not due a separate soul-like entity.
Therefore, it is completely impossible to determine via experiment if ghosts exist. We don't know what they should be, how and if we should be able to measure them, and there are some strong indications that they would be very different from what common definitions would make them.
As a skeptic, this should make your alarm bells start ringing like crazy! We are clearly in a Russell's teapot situation where something is so vague that it is impossible to disprove, however it is clear that it is scientific nonsense.
The best we can do is to debunk specific claims of ghost appearances.
1 See also: