The list below contains some of the places that dairy likes to hide. It is by no means exhaustive and is only a suggestion of where to look for dairy in your diet. (I avoid dairy completely, but depending on your reason for avoiding dairy (lactose free, vegan, casein free) you may not need this entire list.) Also, I find that quite a few people don't actually know what is and isn't dairy. I find it easier to say 'milk' or 'milk products' and then list them out: cheese, butter, sour cream, yoghurt, cream.
I will update this list whenever I encounter suggestions of other places dairy can be.
Mashed Potatoes - made with milk and butter.
Scrambled Eggs - some restaurants add milk.
Grills - A lot of steakhouses (and others) butter their grill.
Rice - most non-ethnic restaurants add butter to the entire pot of rice while it is cooking.
Vegetables - these can either be cooked with butter, or served with butter so be sure to ask.
Margarine - a number of margarines contain dairy. I usually avoid this when eating out as I never trust the waitstaff to be sure.
Soups - even non-cream soups can have a cream or milk base to them.
Gravies - just like soups they can be based on a cream or milk base.
Meatballs - a lot of chefs add Parmesan cheese.
High end restaurants - I find the nicer the restaurant the more likely they are to cook with butter and have contaminated every dish.
Demi glace - often this has a butter base.
Ghee - found in Indian cooking, this is clarified butter.
Bread Crumbs - quite often these have Parmesan cheese as well.
Baked Goods - most pies, tarts, cakes, cookies, etc all have dairy in them.
At the supermarket:
Any non-whole food - dairy can be added to just about any processed food and I've found it in some very odd places.
Bullion Cubes - a couple of brands have milk.
Butter flavor - almost all butter flavor is derived from . . . butter.
Whey - this is the liquid left when making cheese from milk.
Casein - this is the protein in milk.
Lactose - this is the sugar in milk (and most commonly what people can not tolerate).
Flavored Chips - Tostito's Hint of Lime chips contain milk powder. I have no idea why!
Generic Over the Counter Medicines - no idea why but most generics contain lactose while the name brands do not. Oh, and in the United States I've rarely encountered actual medications with lactose in them but it seems to be common overseas.
Sherbet - Sorbet is usually dairy free, Sherbet is usually not dairy free.
Non-dairy Cheeses - Yes, you read that correctly. Casein is apparently what makes cheese melt so well so a large number of alternative 'cheese' manufacturers add casein.