It seems that there is a strong consensus when it comes to Omega 3 fatty acids (such as fish oil) and omega 6 (such as sunflower oil). The first are considered anti-inflammatory, while the latter are inflammatory. Omega 9 fatty acids (such as contained in olive oil) are also largely considered anti-inflammatory, although there are divergent views on whether you can cook with it or not. For example, in the GAPS diet, olive oil cooking is prohibited. In my opinion, as I come from a Mediterranean tradition, it can be used for cooking, as long as it is not subjected to high temperatures, and that it does not reach the smoking point.
But this controversy is more important when talking about saturated fats.

Arguments against saturated fats

The American Heart Association says in a post on May 30, 2014 that “Eating foods that contain saturated fats raises the level of cholesterol in your blood.  High levels of blood cholesterol increase your risk of heart disease and stroke.  Be aware, too, that many foods high in saturated fats are also high in cholesterol – which raises your blood cholesterol even higher”. (8)
Dr. Galland says that “the impact of being a meat eater on your risk of coronary artery disease is pretty well established, it’s because of the saturated fat that raises cholesterol.” Also, “a high fat diet increases the flow of bile. Bile alters the gut micro flora because there are organisms that are killed by bile and bile acids and bile salts, and there are others that thrive in bile” (1). So Dr. Galland thinks that meat fat create a dysbiosis (unbalance in gut flora) and increases the risk of cardiovascular diseases.
Several studies have demonstrated that saturated fatty acids (SFAs) stimulate adipose tissue inflammation by a process that involves Toll-like receptor 4 (TLR4). SFAs also stimulate inflammatory molecules in macrophages. (3), (9)
Chang and his colleagues traced how saturated fats, particularly those from dairy, which are also present in many baked goods and processed foods, can change the composition of naturally harmless bacteria communities in the gut. As the balance of species shifts, it can trigger an immune response that results in inflammation and tissue damage. (10), (11).

Arguments in favour of saturated fats

Numerous doctors have changed their minds about saturated fats over time, and are currently in favor, as Dr Weil wrote in 2011: “You're correct that my thinking on saturated fat has evolved. One catalyst was a scientific analysis of 21 earlier studies, which showed "no significant evidence" that saturated fat in the diet is associated with an increased risk of coronary heart disease.” He is referring to studies such as one that was published in 2010 in the American Journal of Clinical Nutrition (7), where it is concluded that “A meta-analysis of prospective epidemiologic studies showed that there is no significant evidence for concluding that dietary saturated fat is associated with an increased risk of CHD or CVD”
Now, a new study (4) indicates that a diet low in carbohydrates is also more effective than a diet low in fat in reducing  in the blood and reducing markers of inflammation.
One possible explanation for this change of opinion is the consideration that not all animal fats are equal. Animal fats are more inflammatory when the animal has been fed with grain, so that the amount of omega-6 is more important. However, the fat of grass-fed animals contain other anti-inflammatory substances. (6)
Typically, in the studies that indicate an inflammatory effect of saturated fats, the type of food that is fed to the animals which the proteins used on in the study is not specified.
The laboratory studies often use refined products, not natural products. In studies where it is concluded that saturated fat is unhealthy, as published in Nature (10), “milk fat” is used, “a powdered substance that remains when fat has been separated from butter and dehydrated“. This fat is very different from the fat that exists naturally in milk, yogurt, or butter made from raw milk. The health effects of these fats can also be very different.

The cholesterol myth

The myth that high cholesterol foods cause heart disease is being rejected by numerous scientific studies. First of all, 75% of the body's cholesterol is synthesized in the liver, so only 25% comes from food intake. So there is not a direct correlation between dietary cholesterol and blood cholesterol. In fact, a study conducted in 1948, discovered that the correlation was actually reversed, meaning that, the more saturated fat one ate, the more cholesterol one ate, the more calories one ate, the lower the person's serum cholesterol. weighed the least and were the most physically active. (13)
By contrast, it has been shown that diets low in saturated fat are link with increased mortality involving cancer, suicide, cerebral haemorrhage and violence. (14) 
Dr. Weston Price studied indigenous civilisations that maintained good health and high longevity, and found that the diet was very different. Some of them had fish as staple food, others were hunters, others were vegetarians with diary products, etc. But they had all in common a very high fat intake.(15)
Dr. Mercola says: “Remember that 75 percent of your cholesterol is produced by your liver, which is influenced by your insulin levels. Therefore, optimising the insulin level, cholesterol will be automatically optimise and risk of both diabetes and heart disease will be reduced. There is no magic pill to cure heart disease, as the underlying cause is insulin resistance caused by eating too many sugars, grains and especially fructose.” “Elevated triglycerides in the blood have been positively linked to proneness to heart disease, but these triglycerides do not come directly from dietary fats; they are made in the liver from any excess sugars that have not been used for energy” (12)

Animal and vegetable saturated fats

The terms saturated fat and animal fat are often confused. All fats and oils, whether of vegetable or animal origin, are some combination of saturated fatty acids, monounsaturated fatty acids and polyunsaturated linoleic acid and linolenic acid. In general, animal fats such as butter, lard and tallow contain about 40-60% saturated fat and are solid at room temperature. They also contain 30-50% of monounsaturated fat and the rest is polyunsaturated. Interestingly, it is precisely the composition most similar to human milk fat: 48% saturated, 33% monounsaturated and 16% polyunsaturated, and this proportion is the human need for life in our food supply. (12) (2)
Coconut oil, for example, is 92% saturated. These fats are liquid in the tropics but hard as butter in northern climes.  Of particular interest is lauric acid, found in large quantities in both coconut oil and in mother's milk. This fatty acid has strong anti-fungal and antimicrobial properties.
Fats and oils harmful to health are those who have undergone the following processes:
​1 -. Vegetable oil extraction in processes that require heating, or using chemicals.