I would say yes and no. That which most people commonly think of when they encounter the term 'spirit', those sort of spirits certainly do not exist! However in the original and deeper meaning of the term 'spirit', they certainly do exist.
The term 'spirit' arose within the context of mystic discourses. It has over time been heavily misinterpreted due to Naive Realism ( http://bit.ly/duT3V9 ) and then co-opted for political purposes or for fantasy and entertainment. It is these misconceptions that have pervaded our culture and which most people associate with the term 'spirit'.
To understand the mystic meaning of the term requires a radical paradigm shift. If your mind firmly believes that the personality complex within the mind is in fact a physical person and that the mind's stream of experiences is in fact a physical world then it is impossible to comprehend the mystic meaning of 'spirit'. From this world-view the term can only be known as some vague "fairy tale" concept that one either believes or disbelieves.
One must overcome naïve realism to some extent in order to comprehend spirit. To understand the nature of the paradigm shift required please refer to John Ringland's answer to What is consciousness? and also John Ringland's answer to The Big Philosophical Questions: Is the Universe a Simulation?
Within the context of the paradigm described in those answers I can define spirit as an information process. The seemingly 'physical' universe is in fact a virtual construct being animated by information processes. These processes operate within / behind / beyond the virtual world permeating it and animating it into existence. Thus all that we are, that we experience, that believe and that we can be is mediated by the underlying information processes (the spirit world).
An equivalent term from physics would be 'wavefunction'. It is an information process that is non-physical and non-local. It is an un-manifest phenomenon that underlies and animates all manifest forms.
So in a mystic sense, spirits most certainly do exist. In a computational sense the information processes that animate our universe certainly do exist. In a quantum sense the wavefunctions that are the non-physical reality underlying the physical universe certainly do exist.
A lot of what is written about spirits is just misconceptions and fairy tales. However if you turn to some coherent sources, then keeping this understanding of 'spirits' in mind you may find that what was previously just "fairy tale" nonsense is actually saying something quite valid and meaningful.
Such writings are not talking about spirits as if they are 'people' that live on a higher plane of existence to us mortals (that is just a metaphorical narrative). When the symbolic, mythic and conceptual languages are properly deciphered, these writings are talking about primal forces operating within the field of existence, which are an intrinsic aspect of the ongoing process of the real.
BTW John Ringland's answer to What use is the I Ching? contains some talk about spirits in the mystic sense. If interpreted through a naïve realist world-view it would seem spurious, yet if interpreted with mystic insight it makes a great deal of sense.