It depends on the severity of the washing and how often. I will give some examples with the recommended paint. Pick the one that closely matches your situation.
1) Severe frequent washing – This is areas, like commercial kitchens, requiring the use of TSP or harsh cleaners. The best in this situation is a 2 part epoxy.
2) Regular kitchens or bathrooms needing weekly cleaning with household cleaners – A good 100% acrylic semi-gloss will stand up well. A tougher alternative is a semi-gloss (or gloss) oil based enamel. The acrylic paint will require more frequent repainting, maybe every 5-7 years, but is easier to work with and doesn't smell bad. The oil base is tougher, but more expensive and difficult to use.
3) Living spaces and hallways that need frequent spot cleaning, dirt and grime left behind from little fingers – A good acrylic with a satin sheen is a good choice. Once again, a satin oil base enamel is tougher.
I reserve epoxy paint for special uses, it is very expensive, and prefer to 100% acrylics whenever possible.