I use the center point only most of the time (with AI Servo and Back Button Focusing).
Occasionally I'll use Expansion Points or Zone Focus. These are not available on all models (they are on 7D, 5DIII). I never use All Points (Auto Selection). I'll particularly use a multi-point in situations where the subject is against a plain background (birds against a largely blue sky, for example) or a very distant background (miles away). If the subject is against a busy, close background (such as birds with a bunch of trees close behind them), or - even worse - some foreground obstructions I will use a single point only. In these situations using any of the multiple point modes, often the camera might choose to focus somewhere other than the subject, or jump back and forth and not be focused on anything at times.
In general, I prefer to control where the lens is focusing myself.... so I choose a single focus point.
On some cameras, the center point is quite a bit better than the peripheral points. With 7D, the peripheral points are very usable. With 5DII and some of the earlier or less expensive models, the peripheral points just aren't responsive enough for fast moving subjects.
One problem with using the center point only is that your images will tend to all have the subject centered. Sometimes I try to leave a little room for cropping after the fact, so that I can change the composition if I wish.
Another problem with using only one point is that the user has to concentrate on keeping it right where you want focus. It takes some practice and work.... but gives the best keeper rate.
Using Back Button Focusing you can start focus early and keep tracking a subject for a while, before taking a shot (or shots).
In general, USM lenses and large aperture lenses will also give better results.
If possible, stop your lens down a bit to give yourself a little more room for slight focus errors. With birds, if you are focusing on the head and eye and want to keep both wing tips relatively in focus, with longer lenses you are going to need to stop down a bit anyway.
Don't expect perfection. With moving subjects there is nearly always some loss and missed focus. With the right camera and lenses, and plenty of practice, you can get a pretty high keeper rate.... but take extra shots anyway, just to be sure.
