This question is about a specific instance of the measurement problem.
Let's say we excite a single emitter producing a single photon (described by state $ |\psi_{i}>$), which then "travels" through the experimental setup until it can interact with a photon counting detector.
In a simplified picture, the detector can be described by an array of atoms with a ground state and a continuum of excited states. The incident radiation can interact with these atoms and excite electrons to the continuum, which are then used to generate an electronic signal. If its amplitude is above a certain threshold, it will result in a digital signal being sent from the detector to a computer, where the signal can be processed, visualized and seen by a human. (The threshold corresponds to a minimum energy level of the incident radiation.)
With this model, one can calculate the probability to detect a photo-ionization process per unit time: $$w(r,t) = \frac{dP_{e}}{dt} = \xi < \psi_{i}|E^{-}(r,t) E^{+}(r,t)|\psi_{i} >$$
Where $\xi$ is a parameter describing the efficiency of the detector.
With this detection probability density and using the mathematical trick of inserting the identity as a complete set of basis states, one can now define a spatio-temporal wavefunction. (Defining a meaningful wavefunction for the free photon before detection is not straight forward because of its infinite de Broglie wavelength). (The calculations can be found in standard Quantum Optics books)
$$\psi(z,t) = <0|E^{+}(z,t)|\psi_{i}>$$
Going to a single positional variable for simplicity (experimentally for example through the use of a single mode fiber right in front of the detector).
The probability associated with this wavefunction $w(r_{0},t)$ defined above, can be sampled in a typical TCSPC (time correlated single photon counting) measurement.
With each click/measurement of the detector, the state is projected (the wavefunction collapses in the Copenhagen interpretation) and the times between excitation of the emitter and detection of the photons can be plotted. A histogram can be obtained, approximating the probability density $w(r_{0},t)$ (fixed detector position, variable detection time).
The question is now, where does the quantum mechanical "measurement" occur in the detection process. In the sense of projection onto the measured state. Does it happen when the photon ionizes an atom in the detector? This process can be described purely quantum mechanically and projection is not necessary here. The postulated projection could just be an effective model to explain a more complete quantum field theory that has not been discovered yet. where the final state is still probabilistic, however the relevant information about the detector producing a click at time t is determined with unit probability.
Analogous to the process that could happen in a detector is the process of decoherence, which leads to an evolution of the relevant subsystem that is not unitary. (Here the whole system still evolves according to the full Hamiltonian producing unitary evolution.)
The uncertainty is in the exact atomic configuration of the detector about which the physicist does not care. This complicated configuration of the atomic states does however allow for sampling of $w(r_{0},t)$ at different times, otherwise it wouldn't be a detector. If this last part were true, then the measurement problem is still not solved though, because a probability that is almost one does not represent the system going from the "possible" to the "actual" physical realization. For this, the probability has to be exactly one for the realized state. Maybe with a further extension to quantum field theory, it will be possible to achieve this. Then no special interpretation of quantum mechanics will be necessary, because the projection axiom can be dropped completely.
Or does it occur when the experimental physicist looks at his computer screen? This would mean that before he looked at the screen, some electronic bits in the computer representing the arrival time of the photon were neither 0 or 1. The last explanation that only a "conscious being" can project a system onto a quantum mechanical state would imply that quantum mechanics is only a valid model of a reality where these conscious beings exist right now.