What shutter speed to use to minimize hand-held camera shake?

I'm hand-holding my Nikon D3100(18-55mm VR lense) in place where there's plenty of light.

The old rule was to always use a shutter speed equal to or greater than the focal length of the lens being used. So at 55mm a shutter speed of 1/100 sec. Would be sufficient. With the VR (Vibration Reduction) engaged Nikon claims a 3 stop reduction in shutter speed when hand-holding the camera. So in theory you could use a shutter speed as slow as 1/12 sec.

Another consideration is your camera holding technique. If you're trying to hold your D3100 at arms length to compose your picture using the screen on the back of the camera then you really need to watch this video on how to hold a DSLR correctly.
http://www.brentmailphotography.com/free-photography-tips/how-to-hold-your-digital-slr-camera-like-a-pro.html

There's a useful rule of thumb which works for most cameras, which is 1 over the focal length. So at a focal length of 50mm, you need a shutter speed of 1/50 but no lower.

This is not an iron rule, though. A good VR system can let you go a little lower (and a really good one can let you go a lot lower), whereas a camera/lens combination with a high pixel pitch demands a faster speed to avoid the shake.

For something like the 3100, I would try and keep it at 1/focal length or a little faster, assuming you have normal hands in terms of steadyness.

Is your aperture set to f/22 or narrower, else I do not see how that's possible in 'plenty of light'

You don't randomly dial in your shutter speed
Better use it on Auto, till such time you are able to read your light meter and get a working knowledge of the 'exposure triangle'

It there's plenty of light then you shouldn't have to worry. As Sean has said - you can't just change the shutter speed without knowing about exposure and how to meter your shot, unless you are in Auto, or in Shutter Priority mode (which is a semi-automatic exposure mode).

Go read this: Understanding Exposure: http://www.cambridgeincolour.com/...posure.htm where you will learn about the exposure triangle and how the ISO, aperture and shutter speed all relate to each other.

Next read your camera user manual, especially the part about using your cameras internal light meter to measure the exposure.

There's a simple rule which says, if you take the focal length and put a 1 over it, then choose the closest shutter speed to it. So at 55mm, that would be 1/55 - the closest would be 1/50th, or at 18mm that would be 1/18th - the closest would be 1/15th

Unless your hands shake like a 95 year old man's, you have nothing to worry about.
That's the purpose of the VR lens.

I agree with what has been said except for one thing: The rule of thumb regarding shutter speed being 1/focal length is correct, but only for full-frame or 35mm cameras. You must add in the crop factor into the equation. So instead of being able to shoot 1/250th with a 200mm lens, you really should be at 1/340 if your shooting with an APS-C camera (1.6x crop factor), or 1/500 if using a micro 4/3 camera which has a 2x crop factor.

From that point, you can then determine how low you can hand hold the camera based on the effectiveness of your camera's shake reduction system and your steadiness.