We present a major upgrade of the micrOMEGAs dark matter code to compute the abundance of feebly interacting dark matter candidates through the freeze-in mechanism in generic extensions of the Standard Model of particle physics. We develop the necessary formalism in order to solve the freeze-in Boltzmann equations while making as few simplifying assumptions as possible concerning the phase-space distributions of the particles involved in the dark matter production process. We further show that this formalism allows us to treat different freeze-in scenarios and discuss the way it is implemented in the code. We find that, depending on the New Physics scenario under consideration, the effect of a proper treatment of statistics on the predicted dark matter abundance can range from a few percent up to a factor of two, or more. We moreover illustrate the underlying physics, as well as the various novel functionalities of micrOMEGAs, by presenting several example results obtained for different dark matter models.