Method #1

This is useful in selecting eggs when you are in the market.

Fresh eggs, when held to the light, the white will look clear, and the yellow distinct; if not good, they will have a clouded appearance. The shell will be almost transparent.

When eggs are stale, the white will be thin and watery, and the yolk will not be a uniform color, when broken; if there is no mustiness, or disagreeable smell, eggs in this state are not unfit for making cakes, puddings, etc.

From "The National Farmer's & Housekeeper's Cyclopedia," 1888

Method #2

You can use this method at home.

To determine the exact age of eggs, dissolve about four ounces of common salt in a quart of pure water and then immerse the egg. If it be only a day or so old, it will sink to the bottom of the vessel, but if it be three days old it will float in the liquid; if more than five it comes to the surface, and rises above in proportion to its increased age.

From the book, "The Hearthstone," 1887