It is a cornerstone of Quantum Mechanics that you cannot know both the position and momentum of a particle.
What if the particle is a quantum singularity?
A singularity is defined as an area in which the current laws of physics cannot be applied. A quantum singularity does not, therefore, have a defined meaning (nor is it a particle).
The only singularities known to exist in the present Universe are those at the centers of black holes. I would expect them to follow all the laws of quantum mechanics, but being in a black hole, gravity is inherently significant, so you’d need quantum gravity (whatever that is) to fully describe the situation. And besides, all known black holes (and indeed, almost all of the ones even hypothesized to exist) are easily massive enough that the Uncertainty Principle isn’t very interesting.