This article will be a bit dense, as crash courses go :-), but I will summarize at the end for simplified guidelines.
The fats can be broadly classified into 3 types based on their saturation levels. The Saturated Fats (SFA), the Mono-Unsaturated Fats (MUFA), and the Poly-Unsaturated Fats (PUFA). As any chemistry student will tell you the higher the unsaturation the lower the stability of the fat. So PUFA are the most unstable and SFA are the most stable.
A good rule of thumb is to use high heat only on SFA and MUFA fats. PUFA fats should not be heated as far as possible. I am ignoring the fact that all oils/fats that you will buy on the market or get in real life will be a combination of all sorts of fats. I will get back to it later.
SFA fats can be further broken down into 3 types based on the length of the carbon chains. Chains shorter than 8 carbon atoms are called Short chain fats (SCFA). 16 and above are considered to be long chain (LCFA). The ones in the middle are called Medium Chain Fats (MCFA). The three fats have slightly different use in the body. The SCFA are generally used by the digestive system (gut) as food, and much of it gets used there. These are also created by the bacteria in the gut by decomposing fiber. Butyric acid which gives Butter its distinctive smell is a SCFA. The MCFA are broken down into ketones by the liver and are used for energy immediately. Lauric Acid is the major component of Coconut oil and is an MCFA. LCSFA are primarily stored and then used while fasting. Most fats in meat are LCFA, both SFA and MUFA. The LCFA are also used in the body to provide structure. These have a high melting point and form the bulk of cells, neurons, and cholesterol. These are also required for hormone production.
The MUFA are of two types the Long chains and the medium chains. The long chains again can be stored and the medium chains are used immediately for energy. The predominant long chain is Oleic Acid, the major constituent of Olive oil. The Oleic acid is also called Omega9. The medium chain MUFA are not very common. These are as far as I know used primarily for energy. Primarily natural MUFA are cis fats, while artificially created MUFA are generally trans fats. These transfats are solid at room temperature, just like Saturated fats, and are confused with saturated fats. These are extremely harmful for the body, as the body can confuse it with saturated fats, and use these wrongly.
The PUFA are primarily divided into 2 types, Omega3s and Omega6s. Omega3s generally have 3 or more unsaturated bonds while Omega6s are 2 or more unsaturated bonds. So in effect Omega3s are even more unstable than Omega6s. Much of the Omega3s and Omega6s cannot be created in the body (so are called essential fats), although the body can increase the chain length, to create longer chains. PUFA are very unstable. This fact is used by the body, for communication. These are primarily used on the cell walls as receptor sites. These are not used for energy. Some of the Omega3s are used for anti-inflammatory signalling and Omega6s are used for Inflammatory signalling. Some Omega3s are required for brain development, EPA and DHA.
The anti-inflammatory and inflammatory signalling factor means that you want to have a balance of Omega3s and Omega6s. A higher level of Omega6, vis a vis will induce high levels of inflammation. Excess of inflammatory signal can thicken the blood. A higher level of Omega3 will reduce the level of inflammatory signalling, one side effect is that blood clotting stops working effectively. It is best to keep TOTAL Omega3:Omega6 consumption ratio in between 2:1 <-> 1:2. It is generally difficult to get Omega3s from plants. There are a few exceptions like Flax seeds. Be careful with high PUFA oils, you should keep them in the fridge to prevent damage.
The inability to use it for fuel also means that the consumption of PUFA should be very low. There are indications that PUFA consumption should not exceed 4%, ideally it should be at 2% of total energy consumption. This is quite low, around 2gms in a diet of 1800kcal. Also remember you do get some PUFA from your food.
Also remember that Vitamin E keeps the Unsaturated fats stable. So it is best to get some Vitamin E when consuming high PUFA fats.
Our body fat depends on our diet. On a low PUFA diet, it resembles the content of Red Meat, which is about 50/50 division of LCSFA and LCMUFA. The ruminants (eg cow, goat) can convert the PUFA into stable fats so their fat content does not depend much on the PUFA content of the diet. Non-ruminants also have body fat depending on PUFA content in their diet, eg birds, pigs, horses, etc.
Seeds are generally higher in PUFA, than MUFA or SFA, with a very few exceptions, eg Coconuts.
Now that you have a rough idea of what different fats are, you can have a look at this table, for the constituents of different oils. Unfortunately it does not list the animal fats, but it contains a lot of the plant fats.
The C number in the fatty acid composition section, gives you the type of the fatty acid. The table does not list any number below C8, which means that plants fats rarely have any short chain fatty acids. You will also note that they are all in even numbers. There are some with a colon, eg C18:1, C18:2, C18:3. These indicate the degree of desaturation, ie there is 1, 2, or 3 unsaturated bonds. C18:2 is an Omega6 and C18:3 is an Omega3.
Before we get into dissecting each common oils and fats, one word of caution. Fats are essentially nutrient free, except for some fat soluble vitamins. These vitamins are important, but its best to get from whole foods. Added fats should be avoided generally, just like low nutrient grains/legumes should be avoided. There maybe some conditions in which some particular fats can help.
Now lets try to use the above given information on the different common oils.
1) Coconut oil: Around 80% of the fats are Medium chains, and only 20% is long chain. There is some MUFA, but PUFA is very very little. As we know that Medium chains don't get stored, so it would be very difficult to gain weight if Coconut oil is the main calorie source. Also PUFA is very low, so it is the ideal frying oil. It will not go bad easily. At around 1% PUFA, you could have it as much as you want, without overshooting the PUFA budget. You will note that Palm Kernel oil has very similar characteristics. It is IMO ideal for fat loss.
2) Ground nut oil: Around 60% MUFA, 20% PUFA. Wikipedia gives a bigger number for PUFA (32%). Assuming that it has been used a lot in our traditional diets, it should have been low in PUFA. So I would go with the numbers given in this paper. at 20% PUFA it would be a fairly stable fat for cooking, but not frying. The fact that all of the PUFA is Omega6 makes this oil difficult to balance. At 20% 2 tsp will complete your daily Omega6 requirement. Other 1gm of PUFA to be exclusively Omega3s.
3) Mustard oil: 47% Erucic, 14% Omega6, 7% Omega3. The Erucic acid is a MUFA and gives it the pungent smell. At Omega6:Omega3 of 2:1 its fairly balanced. Also at 20% PUFA it would be fairly stable for normal cooking. Again 4tsp would be sufficient for the days allowance.
4) Olive oil: 75% MUFA, < 10% PUFA. The low PUFA content means that it is stable enough for cooking. < 10% of Omega6 means that 6tsp can be consumed safely, with appropriate addition of Omega3 from other sources.
5) Sunflower oil, Safflower oil, Wheat Germ oil, Soybean oil: At >50% PUFA these are best avoided completely. If you want to use keep it in fridge and use with a dropper :-).
6) Sesame oil: At 40% PUFA, it is an unstable oil. It could be used as a seasoning oil, when used rarely.
7) Flax Seed oil: At ~70% PUFA, its a very unstable oil. The only good thing about it is that it can be used to balance Omega6 fats from other foods, as it is high in Omega3 and low in Omega6. The ratio being 3:1 Omega3:Omega6. Not more than 1tsp per day, mostly as seasoning or drunk straight up. Buy only cold pressed and keep it in a dark bottle in the fridge. One thing to remember is that plant Omega3s (ALA) are inferior to those in plant sources, as we get preformed EPA and DHA. The conversion rate in men is pathetic of the order of 1% and 4%, while in women and children its 10% and 30% for DHA and EPA respectively.
For the animal fats lets get the data from other places.
8) Milk Fat aka Ghee: 65% Saturated, with some small chains and some medium chain fats. 30% MUFA and less than 4% PUFA. The composition of PUFA depends on the feed of the ruminant. Grass feds generally have a balance of 2:1 Omega6:Omega3. While non-pastured milk PUFA is almost completely Omega6. The milk fat also contains some natural trans fats called Conjugate Linoleic Acid (CLA) and Vaccenic acid. Vaccenic acid is a precursor of CLA. These are byproducts of the animal trying to convert the PUFA into safer MUFA or SFA, and are at worst harmless. They are sold as supplements though. The < 4% PUFA means that 100 gms can be consumed if pastured and 50gms if unpastured.
9) Ruminant Fat: 43% SFA, 50% MUFA, and 4% PUFA. Nearly same as Milk fat, except, that the chains are longer, and the short chains are missing. The vitamin content is also lower compared to the milk fat. Milk fat has an increased vitamin content to provide for the growth of babies.
10) Fish oil: The fatty acid composition varies a lot across fishes. But still we can note a few things. First Omega3:Omega6 varies from 2:1 to 3:1. The amount of PUFA is higher in fish from colder regions. The PUFA content is generally very high. This means that no more than 1tsp and keep it in a fridge. The other benefits of fish oil is that these have Vitamin A and D.
11) Bird and Pig fat can have as much as 30% PUFA if the diet is rich in soy and grain meal. This happens to animals raised in Confined Animal Feeding Operations (CAFO). Pretty much all of commercialized chicken is CAFO.
Ghee and Coconut oils are the best fats for consumption. Occasional use of Mustard oil and Peanut oil is ok. Olive oil can be used regularly. It is a good idea to have a tea spoon of Fish oil or Flax seed oil for extra omega3s for balancing if not using pastured ghee or animal fat. Ruminant fat is good. Bird and Pig fat is only ok if the feed is not grain based, and low in PUFA. Most seed oils should be avoided.