According to QM a system that has an associated wavefunction in a superposition of eigenstates of a position operator, collapses upon a measurement to a single eigenstate. Depending on the measurement precision in practice the result gives a linear combinations of close eigenstates. However, I don't think it is a measurement problem. The theoretical collapse to a single state doesn't seem to be a good idea.
Basically, when we do a measurement, it involves a designed system which can only be made of materials which are made of atomic structures. In any case, it involves an interaction between two particles that have never definite positions. For example we detect a transition of an electron to a different energy level. Therefore any collapse of a i.e. photon wavefunction is triggered by the interaction of the photon with a massive particle. But the interaction doesn't happen at a definite position. The photon energy quantum is absorbed by the electron but it doesn't happen at a point in space because the electron position is still in a superposition of eigenstates. In this case I would say, the photon wavefunction collapses and the position we get is the superposition of eigenstates of the electron that has absorbed the photon quantum, not a single point in space.