The exact patterns of fingerprints are not inherited; fingerprints are unique to each individual even between identical twins (who have ostensibly identical genes). However, certain significant features of fingerprint patterns are strongly correlated between generations, as sort of a side effect of genetics. The major pattern types of fingerprints are whorls, loops, and arches ( images in this article ). In humans, the foundations of these features form when a fetus is around 10 weeks old. At this stage of development, volar pads (precursors to veins and capillaries) are absorbed into the hands/feet while the skin on the hands/feet begins to fold. The large-scale configuration of these skin folds (which ultimately become the whorls, loops, and arches) depends on the absorption of the volar pads, and that absorption rate is genetically determined. Thus, the overall size and shape of fingerprints as well as the frequency of the major features are inherited. However, the smaller details are determined by the fetal environment. These details (such as exact paths of ridges and furrows and features like ridge "islands", "dots", and "bifurcations", known in the field [ dermatoglyphics ] as minutiae) are determined by the specific surroundings in which the fetus is growing.

Because the density of the amniotic fluid, the position of the fetus, etc. are different for every individual, those minutiae develop differently for everyone, including identical twins.

One answer related to this question is here. We will get more answers by this coming Monday.

Great question. Any individual finger has one of 3 major types of patterns (loop, whorl, or arch). Then there are countless variations that make fingerprints individual. The general type seems to be heritable (able to be inherited). The details (which experts call “minutiae”) are very individual and are probably the result of some random events as the skin layers of our fingers develop. These changes happen during weeks 10 to 15 of fetal development, approximately. So we’re born with the fingerprints we’ll have all of our lives.

The general pattern (loop, whorl, or arch) has to do with the timing of important events in fingertip formation. Since genetics can influence the rate of development, genetics influence which general pattern a finger gets.

Identical twins have the same DNA. They have the same general pattern, but the details of their fingerprints are different.

You can examine your own fingerprints by using an inkpad (like the ones used for crafts) and a piece of paper. A magnifying glass helps a lot. You can see the 3 main patterns on this site:
here. Then you can compare your prints to a friend’s prints and look for more details.

People often say that no two people have identical fingerprints, but is there any way to test whether this is true?

As I understand it, no; they are created by the inside of the mother's womb during pregnancy, which is influenced by random processes. I would guess that the answer is partially yes, but mainly no.

Yes! They are partly inherited!

It's a good science project, to compare your fingerprints with the fingerprints of your family and friends.

"Yes, there is an inheritable quality to fingerprints. Pattern types are often genetically inherited, but the individual details that make a fingerprint unique are not. Humans, as well as apes and monkeys, have so-called friction ridge skin (FRS) covering the surfaces of their hands and feet." Read here.

Fingerprints Science Project.