Find all the (isolated) singular points of $f$, classify them, and find the residue of $f$ at each singular point.
$$f(z) = \frac{z^{1/2}}{z^2 + 1}$$
I think I have $3$ singularities at $z=0,-i$ and $i$ but am unsure about what to do next and what type of singularities they are.
I don't fully understand singularities or residues, can someone explain them to me in this example please.
The square root is not a single-valued function in the complex domain so you have to restrict the domain to, say, the complement of the negative real axis to get an analytic function. Once you do that, the problem becomes well-posed. You should decompose your function into a sum of a pair of functions with denominator $z-i$ and $z+i$ and then the answer falls out.