Suppose I have the following function of a complex variable $$f(z)=\log(z)(z^2+1)^{1/2}.$$ Wolfram Alpha tells me the branch cuts of $f(z)$ are $z\leq 0$ (presumably for the logarithmic term), and $\text{Re}(z)=0$,$|z|\geq 1$ (presumably for the square root), i.e. they are restricted to subsets of either the real or imaginary axes. If I want to use the residue calculus with $f(z)$, my contour had better avoid (i.e. not cross) these branch cuts.
I think I'm right in saying that I can choose my branches to suite my needs. Hence, if it turns out that the Wolfram choice of branch cuts permits an awkward contour, can I make a different choice of branch (by way of defining my branch cuts differently) to make life easier?
For example, instead of the above choice, could I choose my branch cuts to be $z\geq 0$ for the logarithm, and $z\leq -1$ and $z\geq 1$ for the square root? This choice leaves the entire upper and lower half planes "free of problems", whereas the initial choice defines branches along both positive and imaginary axes.
You can certainly put your branch cuts where you want to. The logarithmic term forces a branch cut from $0$ to infinity in whatever direction you desire – it doesn't even have to be a straight line. And for the other factor, a branch cut between $i$ and $-i$ is required. You can take the shorter way, or you can do as Wolfram Alpha suggests and let it pass through infinity.
Of course, this all depends on what you want to do. The application often dictates some special choice of branch cuts. And sometimes, merely using branch cuts isn't sufficient, and you have to use more of the associated Riemann surface – as when you use the Pochhammer contour to study the beta function. There is no choice of branch cuts that will avoid that contour!