For example: I am asked to find the singularities (their location and types) for the function $f(z)=\frac{(z-1)^2\cos^2(z/2)}{(z+1)^3z(z-\pi)}$. I found a pole of order 3 at $z=-1$, a pole of order 1 at $z=0$, and a removable singularity at $z=\pi$ using Taylor Series expansion.
I am confused what the location is though. Is it just the $z=$# value? Thanks.
Yes, location just means the $z$-value where there is a singularity. Viewing complex numbers as points in the complex plane motivates this.