Regarding solubility, how are lipids classified?

Fats and oils are hydrophobic molecules, i.e., they are non polar and insoluble in water. Lipids in general are molecules with a large non-polar extension and so they are soluble in non polar solvents, such as benzene, ether and chloroform.

There are some amphipathic lipids, i.e., lipids whose molecules have a hydrophilic portion, like the phospholipids, giving them the property of being dragged by water, and a hydrophobic portion (non polar).