You’re asking the wrong question. The pan doesn’t care what kind of oil or fat you put in it; the dish you’re making does. There are two factors to consider: the flavor of the dish and the technique used.
Different parts of the world have their traditional fats or oils, and their cuisines reflect this. When I think of Provence, I think of olive oil; when I think of Normandy, butter. Bacon “grease” makes me think of the US south. A dish’s place of origin can often be revealed by the fat called for, or at least it could until the diet police got involved. For a while there, you’d have thought the entire world cooked with canola oil (which is not a traditional oil, by the way.)
You must take into account both the authenticity of the recipe, and the flavor of the end product. While swaps can often be made – because authenticity shouldn’t be a proscription of a good idea – the flavors need to make sense. Take ratatouille, a true French rendition of which requires olive oil. Make a vegetable mélange of the same ingredients, substituting bacon grease for the olive oil, and you’ll have something pretty tasty. Now imagine it with an aromatic coconut oil. I don’t know about you, but the idea doesn’t work for me.
Technique is the other factor. In a frying pan you can pan-broil, sauté, pan-fry, and shallow-fry. Pan-broiling uses little or no fat, so I’ll skip over that. Just be aware that you don’t necessarily need oil at all to cook in a frying pan! The other three methods differ in temperature and the amount of oil used.
Sauté means “to jump” in French. Use medium-high to high heat (so that the food “jumps” as it hits the pan), just enough fat to lightly coat the pan, and keep the food in motion for even browning. This technique is for quick-cooking foods like thin slices of meat and cut-up vegetables. You’ll want some kind of vegetable oil or clarified butter. Regular butter isn’t suitable, as it will burn. Some suggest using a mix of oil and butter, but I find it’s too easy for the milk solids in the butter to burn to recommend this.
Pan-frying uses more oil, ¼” or so, and slightly lower heat than sautéing, medium to medium-high. This technique is better for bulkier foods like bone-in pork chops or chicken pieces, breaded foods like eggplant parmigiano, and small whole fish.
Shallow-frying, deep-frying’s little brother, means to cook a food in oil which comes up to about half the depth of the food (½” or so.) The food is partially, not completely, submerged in oil. This technique is good for browning foods like fritters, battered shrimp, and fried chicken. Because the oil is deep enough, you can use a thermometer to monitor the heat, which should be somewhere between 350°F and 400°F. Any higher than that, and the food will burn.
So what are the choices for oils/fats? There are lots, but the most common ones are generic vegetable (usually a combination of corn, soy, and/or canola); each of those individual oils; safflower, grapeseed, peanut, sesame (not roasted sesame – that’s for seasoning, not frying), and olive (extra virgin or not); butter (clarified or not); bacon grease; and lard. If you render your own, or want to search them out, duck and goose fat are excellent, flavorful choices. Just check the smoke point and consider the flavor of each before choosing.