"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.
Those are Shanghai Rice Cakes - Nian Gao If you google, you'll find a bunch of recipes for what I think of as the "standard" rice cake dish, with shredded pork and cabbage. In most of them, the soaked/softened rice cakes are added at the end, after all the other ingredients, so they only get lightly fried.
Best Answer
Oils and other fats can go rancid if old and not properly stored. Cooking oil should be kept in a cool, dry place with minimal exposure to air. In my experience, Canola oil goes rancid faster than many other oils.
One quick way to see -- smell your oil from the bottle. You could also try heating a small amount in a pan, dipping bread in it, and tasting it. It's best to isolate your components when searching for off flavors.