The only oils that are advocated by OMS are extra virgin olive oil (EVOO) and flaxseed oil.
Flaxseed oil must NEVER be heated, but can be poured onto cooked food.
Cooking with EVOO is permissible but the temperature should be kept low.
When 'frying' food, it is best to do it in water, rather than oil. For example, if a recipe calls for onions to be softened in a pan with oil, just substitute the oil for water. All the oil is really doing in this situation is preventing the onions from sticking - something that can be achieved with water, although you may have to add water as you go along.
Certain oils can be high in saturated fat, and heating oils can drastically change their properties which is why they're not permitted. I hope this brief explanation helps. It is an area that many people find confusing at the beginning. As you continue with OMS, you may find that you can eliminate oil from your cooking methods. I only use EVOO when making white sauce, and occasionally in baking.