Let consider a function f (z) of a complex variable z, which is meromorphic on a finite domain ­ ½ C so that the point 0 2 ­. The finite values fj := f (zj) of the function f (z) are known on the set fzjg, where the points zj belong to a simple closed contour ¡ ½ ­, that contains the point z = 0. We admit that points zj form a dense set on ¡. The function f (z) has a finite number of singular points of polar type on the domain ­, but their number and locations are not known. Also, on the contour ¡ the function f (z) can have both poles and jump discontinuity points (which can be considered as removable singularities). If the function f (z) has poles on the contour ¡, in order to avoid the computation difficulties, we consider that the values of the function f (z) at the points zj 2 ¡½ are given (here ¡½ represents a small perturbation of the contour ¡ ). We aim to determine the locations of the singular points on ­, in particular those on the contour ¡. In our earlier works we have already examined the Pad´e approximation with Laurent polynomials [1] and the Pad´e approximation with Faber polynomials [2]. In both cases we have considered the meromorphic functions on a finite domain of the complex plane with given values at the points of a simple closed contour from this domain. In this paper we examine two algorithms for localizing the singular points of meromorphic function f (z), both are based on the approximation by rational functions. The first one is based on global interpolation and gives the possibility to determine the singular points of the function on a domain ­. The second algorithm, based on piecewise interpolation, establishes the poles and the discontinuity points on the contour ¡.