Does zinc work against the common cold?

Yes. Technically, it does - a recent review of zinc has shown it to be effective in shortening the common cold, but its use was limited by intolerable side effects. Given the common cold will resolve in a few days no matter what you do, trying zinc and risking the side effects might just not be worth it.
No. There are several studies that clearly show that zinc is not helpful in the treatment of the common cold (although one recent study slightly points otherwise). Additionally zinc can be poorly tolerated in tablet or lozenge form causing nausea. A nasal spray form of zinc, sold as zicam, has actually been shown to permenantly damage the sense of smell. Do not use zinc as treatment for colds.