In this paper, a multiscale monogenic analysis is applied to 2D interference fringe patterns. The monogenic signal was originally developed as a 2D generalization of the well-known analytic signal in the 1D case. The analytic and monogenic tools are both useful to extract phase information, which can then be directly linked with physical quantities. Previous studies have already shown the interest in the monogenic signal in the field of interferometry. This paper presents theoretical and numerical illustrations of the connection between the physical phase information and the phase estimated with the monogenic tool. More specifically, the ideal case of pure cosine waves is deeply studied, and then the complexity of the fringe patterns is progressively increased. One important weakness of the monogenic transform is its singularity at the null frequency, which makes the phase estimations of low-frequency fringes diverge. Moreover, the monogenic transform is originally designed for narrowband signals, and encounters difficulties when dealing with noised signals. These problems can be bypassed by performing a multiscale analysis based on the monogenic wavelet transform. Moreover, this paper proposes a simple strategy to combine the information extracted at different scales in order to get a better estimation of the phase. The numerical tests (synthetic and real signals) show how this approach provides a finer extraction of the geometrical structure of the fringe patterns.