Giotto,
Ask and ye shall receive

.
Usually I try to get the best and freshest cheeses I can, whether it is mozzarella cheese (like buffalo mozzarella or a freshly-made or artisanal cow's mozzarella cheese), specialty cheeses (like Fontina, asiago, or Swiss), or hard aged cheeses (like Parmigiano-Reggiano, Romano and grana padano cheeses). Since different pizzas call for different cheeses, I mix and match the cheeses as called for in recipes or as dictated by my imagination. Sometimes I am left with primarily processed cheeses, particularly mozzarella cheeses. Even then, I tend to work down the mozzarella food chain, trying to find the best form of the cheese and combine it with other cheeses to produce the desired end result.
As most experienced pizza makers know, processed mozzarella cheese come in many varieties, including full-fat whole milk (100%), part skim, and low-moisture/part-skim, each with its own qualities and attributes. They are often combined together on pizzas (e.g., a 50/50 blend of whole milk mozzarella cheese and part-skim mozzarella cheese) or used with other types of cheeses, of which a mozzarella cheese/provolone cheese combination is one of the most popular (and one of my favorites) because the two cheeses have similar melting, flow and stretching characteristics (they are both stringy curd cheeses). The provolone cheese (regular or smoked) also adds a distinctive flavor.
While I haven't yet tried it or confirmed it, I understand that a popular cheese combination in the St. Louis area uses provolone cheese, white cheddar cheese and Swiss cheese. Other possibilities for combination with mozzarella cheese are Monterey Jack and orange or yellow cheddar cheeses. Where a buttery-rich flavor and good meltability is desired, I use whole milk mozzarella cheese. However, unlike its lower-fat cousins, it is subject to breaking down and releasing some of its fat during baking, producing an oily appearance which may not be visually pleasing (although New Yorkers love it, especially when the oil runs down their arm while eating

). The oiliness is one of the reasons why many professional pizza makers use whole milk mozzarella cheese in combination with part-skim mozzarella cheese or provolone cheese. I avoid non-fat and imitation mozzarella cheeses altogether because they are the most inferior of all the mozzarella cheeses and there are much better choices from a quality and taste standpoint.
As between the various processed forms, I prefer mozzarella cheese that is freshly sliced from a large block or brick of cheese, usually at the deli counters of supermarkets and food specialty stores. If I could get the Grande cheese in that form, that would be my first choice. However, I have noticed that the "generic" mozzarella cheeses sold at the deli counter often have good parentage--some of the better known brands available. If the deli mozzarella cheese is not available for any reason, then I use the packaged mozzarella balls, of which three well-known brands, the Polly-O (a Kraft Foods product), Stella (which also comes in a deli style and in a coarsely shredded form) and Calabro brands, appear to be among the better ones in that category that I have been able to find. Pre-packaged mozzarella slices are also acceptable provided they are free of chemical additives. As much as possible, or unless I have no other choice, I avoid the very finely shredded, finely diced, or minced forms of mozzarella cheese (which usually come in plastic bags) since the cheese tends to cook too quickly and brown prematurely when used on pizzas baked at high oven temperatures. They are also highly processed with a multitude of additives to prevent caking, inhibit mold and prolong shelf life (usually measured in weeks).
I tend to distribute meltable cheeses like mozzarella and provolone cheeses in small pieces or chunks, large dice or shreds, or as thin slices over a pizza rather than using a finely shredded, diced or minced form (although the roughly shredded versions are an improvement over the other shredded forms). In these forms, the pieces or slices of meltable cheese will melt in "puddles" and stay soft and chewy, rather than turning brown. Alternatively, I withhold the meltable cheeses until a couple of minutes or so before the pizza is completely baked-- something I quite frequently do with Margherita pizzas and other pizzas with few toppings--or a pizza round can be baked without toppings until it sets, and then add the toppings, including the cheeses, and finish baking. If I suspect a particular cheese I plan to use may be prone to premature browning, I will either use thicker slices or put it in the refrigerator to keep it cool and take it out of the refrigerator at the last minute just as I am ready to use it. Sometimes I will freeze the cheese for a few minutes to get it cold.
To the extent that other cheeses are available for use on pizzas, such as fresh ricotta cheese or goat cheese, I take advantage of that freshness. I add such cheeses to pizzas by teaspoon or tablespoonful. These cheeses tend not to have similar melting characteristics as the other cheeses described and, thus, will tend to stay in place on pizzas and not flow into the other toppings used. Since ricotta and goat cheeses are soft cheeses, they can also be combined with herbs or garlic (or bought as such) and lend additional flavor components to pizzas. They should be used somewhat sparingly, however, so as not to overtake other flavors of the pizza (unless the pizza is built primarily around such cheeses).
As for the hard aged cheeses, like the "Big Three" Italian cheeses mentioned above, I prefer to grate them freshly and add them to the finished pizzas. This is to maintain the sharpness and freshness of their taste. Otherwise, they tend to "disappear" into the pizza. I have also used one or more of the Big Three cheeses in sauces, and even in the dough itself (but in small quantity so as not to interfere with the various chemical and biological activities that take place within the dough).
All of the cheeses mentioned above are dairy products. It is also possible to use a non-dairy, soy-based "mozzarella" on pizzas. I have done this on several occasions with surprisingly good results considering that it is not dairy based. Soy-based mozzarella is a firm, mild (but pleasant) tasting, vegetable form of mozzarella cheese made principally from soybeans. It looks very much like regular mozzarella cheese, shreds and slices just like regular mozzarella cheese, and can be used on a pizza just like regular mozzarella cheese. It will melt without any significant browning and it will be chewy and almost indistinguishable on a baked pizza from regular mozzarella cheese. However, it will not be as flavorful as regular mozzarella cheese (although some will argue that it should have a neutral taste and serve a secondary role to the sauce and other toppings), and certainly not as tasty or flavorful as fresh mozzarella cheese. But it is lactose- and cholesterol-free, so it offers clear advantages to persons who are lactose intolerant or are on low-fat or low-cholesterol diets. Combined with a high-quality pizza crust and toppings, the soy mozzarella cheese allows those on restricted diets to be able to enjoy pizza along with everyone else. While I have never done this before, sometime I would like to make and serve a pizza using the soy mozzarella cheese to friends or family just to see if they can tell the difference. Soy mozzarella cheese tends to be sold at health food and organic food stores, such as Whole Foods and Wild Oats, which is where I have found it.
Peter