Shooting at high ISO can yield usable images if shot under the right condition. This is my favorite recipe for a successful high ISO photography using available light:

• Shoot wide open
  • Set the camera to the largest f-stop of the lens.
  • Remove filters. They don't help in low light and can only cause flare.
• Get an accurate exposure
  • Definitely do not underexpose. It will only aggravate the already "noisy" shooting situation.
  • Do not also overexpose. It is a waste of ISO.
  • Use a handheld meter, if possible. Unfortunately, most camera metering gives inaccurate exposure in low light scenes. 
• Set the correct white balance
  • A wrong white balance will lead to color shifts to can affect proper exposure.
  • Use a "white balance" card and set it close to the most dominant light source for the image you want to take.
    • Note that the exposure close to the light source may be different from the proper exposure for the image.
    • To set the WB correctly, you must use the correct exposure.
• Shoot RAW
  • Post processing is a must for optimum image.
  • You get the most latitude for post processing if you shoot RAW.
• Manage noise
  • Noise is best managed outside the camera during post processing. I use DxO Optics Pro.
  • I like my low light images to have visible grains so I replace the digital noise with film grains.
  • To do this,
    • I use DxO noise reduction to remove as much of the noise as possible (without loosing details).
    • Then, I introduce grain into the image using film-rendering software. I use DxO Film Pack.
• Convert to black-and-white
  • My preferred output for most low light images is black-and-white.
  • Conversion to BW instantly solves the problem of unwanted color casts on skin tones caused by artificial lights.
  • It also allows me to use "color filters" to change tones in the my image.

The following images were taken during a gathering at Riddhi's house in Clifton, NJ. All shots taken with a Leica M9 and a 50mm Summicron at ISO 1600 wide open at f/2. For moving subjects (people especially children), I would set the shutter speed to at least 1/45s to 1/60s.