oil for cooking eggs
Assuming you are talking about fried eggs, use a non-stick pan, and use just enough fat to coat the pan. If you're using an oil, don't measure, just use a few drops, and make sure to spread it thinly on the pan. You could also use cooking spray. The key is having a really, really slick non-stick pan. No pan with any kind of texture will do.

For scrambled, the same is true, but I'd suggest using butter, and just a little more than you need. For 4 eggs, I'd use about a teaspoon of butter. Of course, I also add whipping cream to my scrambled, do it over a double-boiler, and often finish with a small amount of melted butter on top as it is. Here, you want to really keep the eggs moving to keep from sticking.

If you're talking about another cooking method, please be a little more specific.