Since $Q=\{y\mid y=\cos x,x\in R\}$,
we know that $Q=\{y\mid -1\leqslant y\leqslant 1\}$.
Also, given that $P=\{-1,0,1\}$,
we have $P\cap Q=\{-1,0,1\}$.
Thus, the answer is A.
First, simplify set $Q$ based on the range of the cosine function, and then find the common elements of the two sets using the definition of intersection to obtain $P\cap Q$.
This question primarily tests knowledge of set intersections and their operations, as well as the range of trigonometric functions. It assesses problem-solving abilities and the capacity to simplify and transform problems, making it a basic-level question.

The final answer is $\boxed{A}$.