What oil or fat to use for different purposes?
"Commonly used" depends mostly on the culture, I'd assume.  There's a lot of different oils, so I've organized by use rather than try for a complete list.

Some of the ones that you might find in a "typical American" foodie's kitchen include:

For frying: something with a high smoke point : peanut, sunflower, soy, extra light olive oil

For baking (muffins &amp; cakes): something with a mild flavor : corn, canola, "vegetable", soy

For baking (biscuits &amp; pastry, or greasing a pan) something solid at room temp : butter, shortening, lard

For general pan cooking: olive oil (any kind), butter, anything from the "baking (muffins)" list.  update: this assumes sautéing heat or lower; see 'frying' for higher heat applications.

For salad dressing: any nut oil, mild oil, or virgin / extra virgin olive oil

For sauces: Butter.

For finishing: something flavorful to drizzle over at the last second... odds are, it's extra virgin olive oil, but possibly sesame or a nut oil.

Now, there's regional differences -- in the south, it's pretty common to save your bacon grease for cooking and to use shortening for frying.  Lard's still popular in hispanic (and likely other) cuisine, schmaltz (rendered poultry fat) is used in both Jewish and French cooking.  Ghee (similar to clarified butter), is used Indian cuisine ... and the list goes on.

If you're looking for a 'must keep on hand' list -- a mild oil, extra virgin olive oil and butter will get you through most anything.  Add shortening if you like baking, and sesame oil if you like to cook asian food, and you'll be prepared for most anything.