"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.
Not much is out there.
Food-grade silicone oil (dimethylpolysiloxane, for the chemists out there) is routinely used in medical and food-prep devices, and it has been approved by the US FDA Office of Food Additive Safety for use as a direct additive in diverse foods, like milk, dry gelatin dessert mix, canned pineapple juice, and even salt. Of course, we're talking about minuscule amounts, in the range of 100 ppm; at high concentrations, it is a skin and eye irritant. So what level is safe? We can't assume that the FDA ever tested it, as the Office of Food Additive Safety is woefully underfunded.
An informal literature search yielded me only a handful of scientific articles looking at silicone oil in frying. The most promising was by Bertrand Matthaus, Norbert Haase, and Klaus Vossman, "Factors affecting the concentration of acrylamide during deep fat frying of potatoes," Eur. J. Of Lipid Sci. & Tech., V.106(11), pp. 793-801 (Nov 2004) (fig. 3 specifically measures acrylamide concentration as a function of the amount of silicone oil). Unfortunately, I don't have a subscription, so I can't tell what the bottom line was.
Myself, I don't see the benefit of using it at home.
Best Answer
For the very best tasting fries, onion rings and battered fish are fried in fat made from rendered beef fat. When I was a cook we rendered down a thousand pounds of beef fat a week, it took days to do. But it made the very best tasting savory deep fried foods. The burning temperature is lowish, so food needs to be cooked at 325 and changed more often. It's not good for things like doughnuts.
I'm not a fan of Canola, I find it often has a taste which I don't like. I don't use peanut mostly due to habit, too many people I've run into with allergies. I can't find rendered beef fat and I'm not going to render it at home.
At home I use a mixture of Sunflower and lard, about 1 part to 3 parts. The sunflower oil makes it easier to handle when filtering through a very fine mesh sieve (more liquid at a lower temperature) after use, then I put it in an old olive oil can which I refrigerate between uses. I leave out overnight before I use it then warm in hot water to pour it out. I don't usually use more then once a month. The more crumbs and batter bits you filter out the longer the fat will remain good. So filter, and don't pour the last bit into the can.
Refrigerated the oil mix doesn't go bad before it becomes no good to use. I get well over a years use. I love sunflower oil, it has almost no taste at all and a high smoke temp. It is pricey.