Spray WD-40 lubricant in all of the moving parts and tightening clamps. WD-40 is a very slippery silicone lubricant commonly used on household products to prevent squeaking and rusting. Just make sure you don't get it on a microphone held by the stand and always wipe off any excess lubricant to avoid dropping the stand when moving it.
Place the microphone stand on a thick rubber mat to prevent sound from the ground or surrounding amplifiers from rattling it. This is called decoupling. This process is commonly used with studio monitors and stereo speakers to reduce the transmission of sound between the speaker and the surface it is resting on.
Replace the boom arm of the microphone stand if you are using high-quality boom microphone stands. Generally the boom arm is the part of the microphone stand that will be making noise, so replacing that may be a cheaper alternative than buying a whole new microphone stand. Replacement boom arms should be available from the retailer who sold you the microphone stand. High-quality, professional-grade boom arms can be ordered from both Atlas Sound and Gotham Audio as well.
Replace regular microphone clips with thick rubber microphone clips. Rubber microphone clips take the decoupling idea even further, and they will not make noise when they start to wear out. Rubber microphone clips are widely available for just a few dollars more than standard microphone clips.
Use a shock mount with particularly sensitive microphones like large-diaphragm condensers. A shock mount is basically a microphone holder that suspends the microphone from the microphone stand. Since these microphones are sensitive enough to pick up floor and microphone stand noise, a shock mount is a wise investment. Shock mounts come with many high-end microphones, but they can also be ordered from most retail music stores.