This is a great question, but unfortunately one that does not have a solid answer.
Fats can have differing chemical makeups, depending on the types of fatty acids they're composed of and whether any other substances (like water) are a part of the mix. For example, a stick of butter like you might have in your refrigerator will have a little bit of water or whey left from the process of churning milk to separate butterfat. If you try to freeze a stick of butter, it will work, but in a two-step process. First, any remaining water in the stick of butter will freeze solid at about 32F or 0C. At this point, the butter will not seem entirely frozen solid and can be pierced with a knife, if with some difficulty. At a much colder temperature (which varies depending on the makeup of the fatty acids in the butter), the soft butterfat will freeze solid like ice. This second step involves the structure of the fatty molecules shifting from a looser liquid to a rigid, crystallized state. Freezing and thawing butter can result in a grainy or unpleasant texture because there are both crystals of water (or whey) and fats breaking up in the process.
Some people will refer to the temperature at which fats and oils solidify as the freezing point, but this is really just the first step in freezing. Oils and fats have much slower freezing processes than water because of their molecular makeup. Fat molecules, unlike water molecules, are not polar, so their attraction to each other is much weaker and causes them to move about (and crystallize) more slowly. If you have ever worked with fats or oils that have been in the refrigerator, you may find that they are solid and opaque, but still spreadable. Because the fat is spreadable, it's not quite frozen yet.
In recent years, many people have turned to a process called cryolipolysis or "fat freezing" to remove unwanted fat from the body. The idea behind this is that chilling body fat to a temperature which initiates the freezing process of fat but does not damage skin is ideal for breaking down fatty tissue in the body. The temperature range for such procedures is typically 23F/-5C to -4F/-20C. From this, we can infer that human fat freezes in this temperature range. The range may account for water content in the body and resistance to thermal change in the tissues due to body heat.