Do we really need an article on this I hear you say. Doesn't everybody already know how to hold a mike! Well the answers are yes we do need an article on it because no, most people haven't got a clue how to hold a mike. (I think that was a Ruddism)
To hold a mike is very simple. Just grab it on the shaft and hold it so that the head of the mike is about 1 - 3 cm's from your mouth. Allow the mike lead to fall freely to the floor with no sharp bends and not taught. To hold a mike correctly is very easy, but since so many hold it obviously wrong, this is one of the few times when it's instructive to list the ways in which not to hold it. If you're doing one of these, then its time to make a change.
How to "NOT" hold a microphone
With respect to the distance from your mouth, 1 - 3 cm's is arbitrary. It's just a starting point for most singers. It's the right way to begin most songs as against 30cm's away from your mouth as you might see mikes used by newsreaders or TV reporters. Most microphones used on karaoke stages will not have their gain turned up enough to pick up your voice from 30cm's away (unless of course you're projecting a fair bit) because it would cause audio feedback (horrible screeching sound that damages performances, ears and equipment).
So what is the correct distance then? Well, it depends on the expression you're putting in, but generally, it should be at a distance that you think will sound the best for the audience. The best thing to do is to listen to the sound of your voice as best you can coming from the speakers as opposed to coming from your mouth. Your ability to do that will be limited by the foldback capabilities of the pa system as controlled by your host.
Some songs, have very quiet parts that may be not much more than a whisper and some very loud parts as well. During a whisper you might get under 1cm and then when you're unloading a lot of volume and projecting right out there, 30cm or more might be appropriate. Once the host has set the volume for mike, a good singer will work with that such that they modify their own volume and mike distance to get the right mix of sound.