Do singularities always appear on all Riemann sheets?

Question

Consider a function $$f(z) = \frac{\ln z}{z^2+1}.$$ Besides the branching point $z=0$, the function also has singularities at $z = \pm i$. This singularities should appear on all Riemann sheets.

Is there a function that posseses a singularity on a specific Riemann sheet and not on all of them?

Answer 1

$\log(z)^{-1}$ only has a singularity at $z=1$ for the branch where $\log(1)=0$.