I don't believe in the sterotypical ghost, eg. Jacob Marley or Ghost Whisperer.
Also let me say that I consider myself to be a scientist, first and foremost.
But since I first moved into my present home 16 years ago I've had several first hand experiences that make me wonder.
When I was first moving in, several times I noticed that the light at the bottom of the cellar stairs was on. I'm not one to not turn off light switches unless I plan on going back again soon. I shrugged it off figuring I must just have missed it, new house and all. But after it happened several times, and only with that one particular light, I started getting a bit creeped out. Even when I made a conscious effort to purposely turn it off and remember doing so, I still later found it on again a few times. Faulty wiring? It stopped happening so I forgot about it.
I was using the room at the bottom of the stairs for a "music" room where I played my drum set occasionally. Sometimes while getting into it I swore I heard something outside the door (where the light is). I'd stop playing, nothing. OK reverberation rattle, no biggie.
Then one day while I was going down stairs I mis-stepped. The top stair was missing the aluminum kick protector on it's edge. Not looking exactly at the stairs, my peripheral vision was able to see the edge protectors on the other steps, but as the top one didn't have one, my mind interpreted the top step as being part of the top landing. So when I put my foot down it went further than I expected. Luckily I use hand rails. At that instant an image flashed into my mind. An elderly woman lying in a heap at the bottom of the stairs under the light.
I've never tried to find out how the past owner of my house died (she was an elderly widow). But I have a feeling I already know. Several days later while taking a short-cut through a cemetary, for some reason I happened to glance at a grave stone. It was her and her husbands.
I eventually shrugged all this off as coincidence. I am a scientist after all.