There are two kinds of stone commonly referred to as alabaster. True alabaster is gypsum, a soft and chalky sulfate of calcium. Oriental alabaster is carbonate of calcium, better known as marble. Most surviving ancient alabaster is the marble variety, while modern work commonly uses the softer stone. According to the National Park Service's conservation guide, alabaster is an especially difficult stone to preserve and maintain. This is because of its unusual softness, porosity and water solubility.
True alabaster is one of the softest of all stones. It can easily be marked with a fingernail, making it painfully vulnerable to accidental damage. This also militates against cleaning its surface by scrubbing, which can easily damage the finish. Even when highly polished, alabaster is porous and will absorb any oils or liquids it comes into contact with. It's also water soluble, which means even a damp cloth can damage its finish.This combination of traits and vulnerabilities makes Egyptian vases a challenge for conservation purposes.
The National Park Service's foremost recommendation for owners of alabaster vases and other artifacts is to consider the aging of the surface as a desirable patina to be left in place, rather than as soil to be removed. If this gradual darkening is unacceptable to you, they suggest using a soft brush to regularly remove dust buildup, which can trap moisture and cause staining. Additionally you can use oil- or wax-based cleansers for the stone, though these can also cause discoloration over time. Museum-quality antique vases, collectibles or objets d'art of high value should be professionally curated.
The marble form of alabaster, often described as Oriental alabaster, is uncommon in modern Egyptian vases. However, it's widely used in a variety of other artifacts including vases in the Egyptian style, so it is useful to know how to maintain this material as well. Although harder than true alabaster, marble is also highly porous and can be stained by rust, oils and many other substances. Unlike true alabaster, marble is not water-soluble, so the surface can be cleaned gently with a soft cloth and clean water. Don't use soap or detergent, which can stain or mar the surface.