"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.
The major difference between butter and oil is that butter is only 80% oil, with the rest being milk solids and water (source). This means that using oil you will lose some of the water content that your cake should have, possibly resulting in a dryer cake. On the plus side, cakes made with oil tend to dry out slower than those made with butter.
The other possible different, but one that is unlikely with a mix I'd think, is that it depends on the method of mixing in your fat. If the butter is to be whipped or creamed while softened this creates a different texture than one you could get with oil or melted butter. Whipped butter or especially butter creamed with sugar provides a network of air bubbles that act as a raising agent during cooking, and the result is a pound cake like crumb. If your recipe calls for melted butter, though, this is not a concern - the result is more like a traditional muffin or quick bread crumb.
Finally, a vegetable oil has a fairly neutral taste. Usually when a vegetable oil is called for, this is desirable. Butter on the other hand has a more interesting flavor. Your cake may be a bit bland without it.
Best Answer
Fatty acids are not the emulsifiers here. Long chain fatty acids are excellent for emulsification, if they are deprotonated. But then they would be called “fatty acid salts”, and their flavor would be soapy - bitter. If they're neutral fatty acids, they're not ionized enough to retain a sphere of water around the micelle, and block aggregation of the droplets of fat in an emulsion.
Note that the term “fatty acid” is often misused. It is used to describe structural components of triglyceride fats, but in that form they are esterified, and not in their acid forms. They only become free fatty acids after the saponification (breakdown) of a fat, which can happen when cooking fats at high temperature. It's convenient to talk about them as fatty acids when they're still inside triglycerides because of their roles in nutrition (saturated vs. unsaturated vs. polyunsaturated, etc.).
An emulsifying agent like lecithin has two non-polar (electrically neutral), lipophilic (fat soluble) “fatty acid” (but still esterified) hydrocarbon chains, and a polar head consisting of a phosphate/choline (-/+) ionic structure that is very hydrophilic (soluble in water) due to having both positive and negative standing charges. Because of its net electrically neutral structure, the flavor is mild and fatty, not soapy. Phospholipids like lecithin are available in all cells, since they form the lipid bilayer that defines each cell.
Neither vinegar nor lemon juice would act as an effective emulsifying agent. Acetic acid, citric acid, etc., are all too water soluble, and wouldn't stick to the surface of the fat droplets in an emulsion. Rather, components of the soy milk, particularly soy lecithin, would fill that role in this recipe for vegan mayonnaise. In traditional mayonnaise, of course, egg yolks are a rich source of lecithin. The vinegar has little effect other than flavor.
The degree of unsaturation in the non-polar tails of a phospholipid would not significantly affect the stability of an emulsion once formed. Unsaturation of fatty acid esters in triglycerides does affect the texture of a fat, and how it varies with temperature.