One thing you may find useful is a tool that allows you to extract the EXIF data from a digital photo. Most digital cameras will store the settings used to take the picture: shutter speed, f-stop, ISO equivalent, focal length, focus distance, and a ton of other stuff.
There are applications which allow you to extract and view the information as a table (handy for when someone, like MarkWildey asks "What shutter speed did you use?"
There are several you can use (and, as in most things, you can spend little or a lot, depending on what "features" you want.
Here's one that's free:
FWIW, to me that image you posted looks like motion blur. It would be easily proven with a tripod (or other reasonably steady support) and a test image (which could be as simple as a black square outline drawn on white paper. Make sure that the lines are all equal width. Better yet would be 2 concentric black squares separated by a thin white line. You can do this with electrical tape on paper. Shoot the square , uh, square, I guess, and on a point so it's diamond shaped, so you can tell if any blur is vertical or angled. )
I have noticed a lot of people do not wait long enough before dropping the camera away from their shooting stance - this is especially true of cameras that use red-eye reduction. All digital point-and-shoot cameras have some degree of shutter lag, and it seems that people expect them to act like film cameras (shutter goes as soon as the button depresses.)
In any event a bit of experimentation will be quite useful in figuring out what's going on.