I'll give an answer to get us started and hopefully others will add to the conversation as I would like to see a well developed answer to this question.
Fwiw, I wouldn't use Iron Chef as an example of what happens in any professional kitchen. It is a television show, set up for entertainment and as such is organized to that end. So while they wear chef coats and use knives on foods, the activity in a pro kitchen is completely different.
Otherwise, I think the answer depends largely on the individual kitchen, with beef and chicken and veal stocks being the most likely to be found in most places that use stocks. I should point out that when I say most pro kitchens, I am including all commercial kitchens, restaurants of all kinds and institutional, not just high end restaurants. Many kitchens depend on prepared bases from companies like Knorr Swiss and Minor instead of making their own on a regular schedule. These products are fine for general purposes and allow for easy access to stocks without the labor.
In traditional european cuisines, veal stock is considered the best general stock for all purpose use as it lends body and richness to many dishes while not contributing any strong flavor of its' own. A light chicken stock is also popular for many savory dishes. Beef and fish stocks are typically stronger and used specifically for beef and fish dishes, generally speaking.
With the current popularity of mixing foods from different cultures-"fusion cuisine"-having a supply of stock on hand is not as important as it used to be. Much is made of various fruit and vegetable purees and juices, pan reductions and other flavor enhancing liquids. A stock after all is simply flavored water. With the inclusion of a more global perspective on what goes in to a good dish, many things can be used to enhance a dish. So a given "pro" kitchen may have a varied and interesting menu without keeping many stocks on hand.