For numerical computation, a power series representation of a function is typically truncated to a polynomial. Being an entire function, a polynomial cannot reveal much of the singularity structure of the underlying function other than perhaps the distance to the nearest singularity in the complex plane. The same remarks apply to Fourier series and truncated Fourier series. In the first part of the talk we survey some numerical strategies for uncovering additional singularity information. This includes methods based on a theorem of Darboux, as well as Padé and Fourier-Padé approximations. We discuss numerical implementations, stability, and pitfalls. Our test examples include meromorphic functions as well as functions with branch point singularities. In the second part of the talk, we apply these techniques to the computation of some special functions and to a nonlinear PDE.