momentum - What is the reason behind why a quantum particle cannot be at rest?

So I've seen different reasonings for this; which is correct, or are they both corollaries of each other?

1) For a particle to be at rest, we would know its momentum and therefore by Heisenberg's uncertainty principle, the uncertainty in its position would be infinite. Is this acceptable?

2) $\lambda = h / p = h / 0 \implies \lambda \rightarrow \infty$ which is not possible.

EDIT:

Another reason:

3) Due to the zero-point energy, even in its ground state there will always be non-zero kinetic energy.