This question has been explored at length, and with some variation of results, in resources available online. The most common consensus appears to be 27 DOF for the human hand. I’ll try to come up with my own count without referring to their methods.

Each finger has 2 knuckles, each providing a single DOF as they do not move in more than one dimension. One could argue that each pair of knuckles, though they can move independently, provide only a single DOF as they move in the same plane, but I will consider them separately. The base of each finger (including the thumb) is a rotational-type joint, each adding 2 DOF. If we consider the wrist as adding another 2 DOF, this sums to 22 DOF. As the joke goes, “Pull my finger”, gleaming the idea that each of our fingers have an additional uncontrolled DOF, bringing the sum up to 27.

The papers Modeling the Constraints of Human Hand Motion and Model-Based Analysis of Hand Posture both say that the “human hand [has] roughly 30 degrees of freedom”, then go on to use 27 degrees in their application. Their vague language confirms that there is debate as to what constitutes a degree of freedom.

Although I also came up with 27, our counts disagree. Lee & Kunii’s paper describes joint movement as one of flexion, twist, directive, or spherical. Each can be seen as a sub-type of Mataric's rotary joint (none are prismatic) (Matarić, p. 62).

Human hand joints can be classified as flexion, twist, directive, or spherical according to the type of movement or possible rotation axes. A flexion movement joint with a single degree of freedom (DOF) is the knee or elbow, while a twist movement joint having one DOF is the pronation of the forearm. Directive movement with two DOF permits flexion movement in more than two directions. Spherical movement, as in the shoulder or hip, has three DOF and permits simultaneous directive and twist movements.

[Figure 3] shows the joints employed in our hand model, along with their associated allowable movement types. Each finger (II – V) has four DOF (two at the metacarpophalangeal or MP, one at the proximal interphalangeal or PIP, and one at the distal interphalangeal or DIP), while the thumb (I) has five DOF (two at the MP, two at the PIP, and one at the DIP). The wrist’s twist movement is included because the hand must be considered separately from the lower arm. According to this joint movement classification, 27 DOF exist for the hand, including six DOF to position and orient it.

Lee & Kunii consider the thumb to have 5 DOF where I counted 4, the wrist to have 3 DOF where I counted 2, and add 4 more to “position and orient” the hand. I concede that the thumb does indeed have one more than I initially thought, having mistaken the directive joint movement of the MP for flexive; and the wrist, too, having mistaken the spherical joint movement for directive. The last 4 position the entire hand, something that is done by the rest of the human body and not the hand (or wrist) itself. They did not count joint distention as I did for each finger and thumb, which is reasonable considering the tiny limit of this movement.

In context of a hand disconnected from a human (let’s imagine a robotic hand instead of more gruesome options) or in context of a person who is immobile save for their hand (and wrist), these last 4 DOF should not be counted. The total depends on what one wishes to achieve in counting the DOF. The authors of Modeling the Constraints of Human Hand Motion (Lin, Wu, & Huang, 2000) had the goal of capturing hand motion and position with a computer vision system, which means they had to consider the position and motion of the whole hand and wrist system as it was moved by the rest of the body. A robotic surgical tool for doing work on a human hand, in contrast, may assume that the rest of the body has been immobilized. (More about the contextual considerations for DOF in blog post here: https://landing.athabascau.ca/blog/view/2691307/considering-context-when-determining-degrees-of-freedom.)

In conclusion, the human hand has 27 degrees of freedom for most applications, 23 if the rest of the body is immobile. Your goal may allow a simpler model and lower DOF count, or require a more complex model and higher count (e.g. consider each bone in the midcarpal joint independently).