I think this question requires some care in answering, because it seems to be framed in a way that references the phenomenon of neutrino oscillations. It was the observation of neutrino oscillations that established that neutrinos must have mass, and hence not travel at the speed of light.
However, the salient point is that with neutrino, they oscillated between different detectable forms. That means that the neutrino has a detectable degree of freedom.
The photon doesn’t have many detectable degrees of freedom. One is the polarisation, another is the wave vector, and another is the mode frequency/energy. None of these degrees of freedom changes with time during photon propagation. Of course we need to add the caveat that changes in the wave vector and mode frequency can occur in curved and expanding spacetime, but these are external effects, not spontaneous internal changes.
One of the strange things about photons is that even though they can be associated with a frequency, they can’t be observed to oscillate. The idea of oscillation in this context must be associated with some observable property changing over time. This is where visualising a photon as some oscillating wave packet can send the wrong message. In quantum physics, the only things of importance are the properties that you can measure. You can measure the polarisation. You can measure the energy/frequency. You can measure the wave vector. However, you cannot measure anything “oscillating”. Therefore, the photon has the properties consistent with a massless particle.
EDIT: This last paragraph seems to confuse a lot of people. In particular, frequency without oscillation. This is a difficult concept, because it is a specifically quantum concept. The photon is a quantum object and cannot be understood in purely classical terms. If something oscillates, it must have a property that changes in time. This property must be measurable. So what can we measure that can show the oscillating nature of the photon? We can define an electric field operator. Classically, the electric field oscillates in time. However, when we construct this operator, we find that the photon state is not an eigenstate of the electric field operator. Furthermore, we can construct a phase operator that should tell us where in the oscillating phase the photon is. However, the photon state is not an eigenstate of the phase operator. That means the phase of a photon is indeterminate. In fact, there are no operators that we can construct that reveal any time varying properties of the photon. Yes this is weird, but that is quantum theory for you. You often have to leave your classical intuition at the door.
The following diagram illustrates the point. It is called the Wigner function of a single photon state. The Wigner function is a quasi-probability function defined in phase space. The axes defining phase space are the orthogonal phase quadratures. You can think of them as position and momentum, because they map dynamics of a simple harmonic oscillator, such as a pendulum. The oscillating pendulum would be a single point on phase space (specifically a Dirac delta function corresponding to unit probability at the point) that traces a circle around the origin over time as the pendulum swings back and forth. The snapshot at a specific time will then just be a point displaced from the origin. The displacement represents the amplitude of the swing, and the angle relative to the axes represents the phase. Thus the pendulum can be said to oscillate as it rotates around the origin over time. When it comes to light, a laser will look just like the case of the pendulum, except the point is replaced by a 2D Gaussian function. If the Gaussian is displaced sufficiently from the origin, it looks just like a point. That's the classical limit. The Gaussian envelope represents the quantum noise on the classical signal. That's called shot noise and is a representation of the Heisenberg uncertainty principle.
Now the following figure is the Wigner function of a single photon Fock state:
Notice two things. First is that the Wigner function is negative at the origin. This is why it is called a quasi-probability function. Negative probabilities have no classical meaning. If they are found in the Wigner function, it is an indication of non-classical quantum behaviour. Therefore expect classical intuition to fail in these cases. This is why the quantum nature of the photon is difficult to describe.
The other point to notice is that the Wigner function is symmetric around the origin. Recall that this function represents a snapshot in time. This is telling us that the phase of the single photon state is indeterminate. This contrasts with classical light, such as that from a laser, that has a very well defined phase. So while laser light can be said to oscillate, a single photon cannot.