Ex. 1:   f(z) = Csc[z];   isolated singularities on real axis

Ex. 2:   f(z) = Sin[z]/z;   removable singularity at z = 0

Ex. 3: f(z) = (Cos[z]-1)/z^4;   pole order 2 at z = 0

Ex. 4: f(z) = (2z+4)/((z+2i)(z-4i)^2 Sin[z]);   various poles

Ex. 5:   f(z) = Exp[1/z];   Essential singularity at 0 shown at different scales