"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 & cakes): something with a mild flavor : corn, canola, "vegetable", soy
For baking (biscuits & 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.
With Olive oils, the more refined they are, the higher their burning point. So you are correct, an Organic Extra Virgin Olive Oil would be a terrible choice for Indian cooking (would cause effect on taste, smell, and nutrition) which has prolonged periods of sauteing on high heat.
Lower quality olive oils, or a light olive oil, interestingly, would be a better choice. They are much more refined like vegetable oils, so have a higher burning point. But at that point, you'd consider why are you using Olive oil?
Consider using refined butter (ghee) or coconut oil for Indian cooking. Ghee and Coconut oil will rarely smoke or burn and can stand high heat pretty well. I believe traditional Indian cooking uses ghee.
Best Answer
I don't want to disappoint you, but the sad truth is that extra virgin olive oil is unsuitable for all the cooking methods you mention. When you heat any oil past its smoking point it starts to deteriorate and can even become dangerous. Olive oil, extra virgin in particular, has a lower smoking point than most other oils. In fact, you will be better off with an olive oil of lesser quality. Such an oil will have been processed and thus purified, increasing its smoking point.
You may think that butter is unhealthy, but at high temperatures it will actually withstand the heat better than extra virgin olive oil.
Good oils for the purposes you mention are canola oil, sunflower oil, peanut oil, and grapeseed oil.