Position of the wavefunction's collapse

When a wave function is said to "collapse" to a single point during a measurement, is there uncertainty about the point's position or is it known infinitely precisely?

• It is known precisely, although in practice one cannot construct experiments that measure to arbitrarily precision. – Charlie May 28 at 16:04

A better understanding is that if you were to make a position measurement of your system, then the wavefunction "collapses" to a state with definite position. This wavefunction is $$0$$ everywhere except at the location where the particle was observed to be at.