This seems an appropriate topic for Halloween. (I'm going to blithely ignore the various origins and implications of Halloween; perhaps those will turn up in a later post.) I've found that its rather hard to be a Christian and utterly deny the existance of ghosts. I haven't seen one, personally, and I may never, but I cannot say they don't exist.
Aside from modern anecdotal evidence, there are plenty of Biblical references to ghosts. When Saul visited a medium to conjure the spirit of Samuel (1 Samuel 28), Scripture states that it was Samuel speaking - not something that took Samuel's appearance. If this was a demon speaking in the guise of Samuel, it seems like the Bible would include that important fact.
When Jesus walks on water (Matt 14:23-26), one of the apostles in the boat believe he's seeing a ghost. This, at least, demonstrates a belief in ghosts at that time. And similarly, when Jesus appears to them in the locked room (Luke 24:36-39), they, at first, believe He is a ghost. Jesus reassures them with the words, "a spirit has not flesh and bones as you see that I have." The catechism of the Catholic Church seems to support this interpretation, noting that "so impossible did the thing seem: they thought they were seeing a ghost."
My favorite piece of Biblical evidence - and the one that takes us from 'maybe' to 'must' on the subject of ghosts - occurs at the Transfiguration (Matt 17 and Mark 9). The apostles offer to set up tents for Moses and Elijah; clearly, they are seeing these two long-dead figures. If they weren't alive, and they weren't bodily resurrected (which occurs at the end), then what else can we call them but ghosts? (And don't say "angels"; people don't become angels... but that's another post.)
We have a rich, interesting, sometimes surprising legacy as Catholics. Somehow, our beliefs - not just about ghosts, but also marriage, abortion, politics, and so on - have been watered down over the years. Remember, Jesus is "not a tame lion"; don't be afraid to belief in "weird things".
I'll end with a quote from G. K. Chesterton that seems particularly relevent:
"Somehow or other an extraordinary idea has arisen that the disbelievers in miracles consider them coldly and fairly... The plain... course is to trust... [a] peasant’s word about... [a] ghost exactly as far as you trust the peasant’s word about the landlord... If you reject it, you... mean one of two things. You reject [it]... because the man is a peasant or because the story is a ghost story. That is, you either deny the main principle of democracy, or you affirm the main principle of materialism – the abstract impossibility of miracle... in that case you are the dogmatist... constrained to do so by your creed. It is we Christians who accept all actual evidence."