
There's an often cited rule in photography that in order to get sharp images when shooting a camera hand held (rather than on a tripod), you have to use a shutter speed which is (1/focal_length) seconds. So for a 100mm lens you'd need to shoot at 1/100s or faster, for a 300mm lens, 1/300s or faster and for a 500mm lens, 1/500s or faster.
Does this rule work, does it apply to all formats and where does it come from?
The answers to the first two questions are "sometimes" and "no", but the third question requires a longer answer - and the answer will also address the first two questions!
When you are handholding a camera, it's never perfectly still. It will be tilting up and down and swinging left and right due to involuntary movements of your muscles. How much it moves depends on a lot of things. The weight of the camera and lens, whether you are standing, kneeling or resting against a doorway and possibly how many cups of coffee you drank that day!
However, let's assume on average that you are swinging the camera though small angles at a rate of, say, 0.8 degree per second. The amount of movement of the image on the sensor will depend on the focal length of the lens. You can easily see this for yourself when you compare the view handholding a 20mm wideangle lens (very steady image) to that when handholding a 500mm lens (the image moves at lot).
For a 100mm lens, a 0.8 degree swing will move the image through about 1.6 mm, which will cause considerable blurring. If you want no significant blurring, you'd want to hold image motion down to something like 16 microns for an 35mm frame (16 microns is about 1/2 the value of the circle of confusion and is equal to 16/1000mm). Since the image moves though 16 microns in 1/100s, you'd want to shoot at a shutter speed of 1/100s or faster.
Now this corresponds to the 1/focal_length rule because I chose the 0.8 degree/sec movement rate so I got the right answer! Obviously the rate at which the camera moves around will vary. It times it will be slower, at times it may be faster, for some people it may be slower, for some people faster. That's why the "rule" is only a guide. Sometimes 1/100s will be fast enough to give a sharp image, sometimes it won't be. Sometimes you may get a sharp image at 1/50s or 1/25s, though the probability of getting a sharp image drops rapidly at slower shutter speeds. However on the average, if the shutter speed is 1/focal_length or faster, you have a good chance of a sharp image.

The set of images above show these effects quite well. They are 100% crops from sets of 5 successive images shot with an EOS 40D using a 100mm lens. As you can see, as the shutter speed drops, the probability of getting a sharp image decreases, but even at 1/160s not all the images are equally sharp. They might all be "OK", certainly for small prints, but they're not all as sharp as they would be if a tripod had been used. The image marked with a red "*" is the only one that's as sharp as a tripod shot.
So now we've answered two of the questions: Where does the 1/focal_length rule come from (see above) and does the rule work (often it does, sometimes it may not). Now let's look at the third question, does format size matter.
Yes it does, and here's why. We judge sharpness based on a print, not based on the image formed on film or a digital sensor. That image must be enlarged to make a print. For 35mm film and full frame sensors, to make an 8"x12" print the original image must be enlarged by a factor of 8 (since the image itself is about 1"x1.5"). However for a Canon APS-C sensor (which is smaller) it must be enlarged by a factor of 12.8x. If we want the same degree of sharpness in the print, the image on the sensor must start out sharper. In fact for Canon APS-C sensors, 1.6x sharper. So instead of allowing the image to move by 16 microns, we can only allow it to move by 10 microns. Since we can only allow 10 microns of image movement, we now need to use a shutter speed of 1/160s rather than 1/100s. So smaller formats need faster shutter speeds if we want equally sharp prints of the same size. Again though, even using a 100mm lens and a shutter speed of 1/160s, you are very unlikely get every shot as sharp as it would be if you used a tripod.
If you don't want to use a tripod, your other alternative is either to use an image stabilized lens if you shoot with Nikon or Canon or to use an image stabilized body if you shoot with Sony, Olympus or Pentax. Either one can give you up to about 3 stops of added stability, so you could get images as sharp at 1/20s with stabilization as you could at 1/160s without it! For more details see the article here on Image Stabilized Lenses and the article on Body and Lens stabilization compared
