Can quantum particle wave functions be located on multiple points in space at once? When physicists say that a quantum particle's position is uncertain, does that mean specifically that the location where the wave function may collapse on a detector is uncertain? 
Or can a wave function itself be centred on multiple points in space at once? 
 A: 
When physicists say that a quantum particle's position is uncertain, does that mean specifically that the location where the wave function may collapse on a detector is uncertain? 

Not only the location, but also that our mathematical wavefunction  models do not predict a specific location for a particle in the quantum mechanical frame of the microcosm. As a rule of thumb there is the Heisenberg Uncertainty principle, HUP.
For the space variables it has to obey $\Delta x \Delta p \geq \hbar/2$ where p is the momentum . So defining an $x$ point in the wavefunction constraints to momentum, introducing a fuzziness, which has been validated with measurements in experiments.

Or can a wave function itself be centred on multiple points in space at once? 

It is not clear what you mean, a wave function $\Psi(x,y,z,t)$ is mathematically completely defined for a specific boundary condition problem. Its modulus squared will give the probability for finding the particle at a specific $(x,y,z,t)$ when measured.
