One of the few ways that a fixed gear bicycle is superior to a freewheel is that it's far far easier to balance on the bike at a full stop (such as when you come to a light and have your toes clipped in). For one thing, you can roll back & forth which helps, but more importantly, you're constantly in tension between your feet and the movement of the tires, which is what allows you to balance. Inspired by this I got an idea for how to balance on a regular freewhile bike like mine so the fixies can't show me up.
The secret is : brake.
As you're coming in to a light to stop, you apply the brakes to slow down, and you should stand up. You will be standing to balance. Your two pedals should be at the same height, eg. horizontal to each other. Now, as you come to a stop - simply keep holding the brakes compressed, and apply forward pressure with your feet. You are now in a tense equilibrium - your feet are gently pushing forward in the pedals and your hands are holding the brakes which keeps the bike still. This is "The Balance". Once you have mastered The Balance you be able to simply hold here and keep the brakes fully held and be completely still.
When you are working on developing The Balance, you should let off the brakes gently so that you can inch forward very slowly. This makes it a lot easier to balance; if you like you can imagine that you're on a Segway - you have your weight to the front of the bike to hold the brakes, which makes you want to topple forward, you compensate that by allowing the base of the bike to slide forward to get under your falling weight and keep you balanced. When working on developing The Balance, you should stop 5-10 feet behind the crosswalk so you can allow yourself to inch forward very slowly while the light is red. Practice by trying to move as little as possible during the red. If you feel yourself losing your balance, let the bike go forward by releasing the brake, don't put your feet down.
This will be much easier with something like Power Grips - some kind of good toe clips; I can't do it without clips yet but I imagine it's possible. The reason you need to be in tension with the brakes on is so that you can use your leg muscles to move your body and support yourself in balance; it turns the pedals into fixed structures that translate your force to the ground. If you don't use the brake trick on a freewheel bicycle, the pedals will either just let you fall or make the bike move.