We have presented the principles of potential-field data decomposition in the Poisson scale-space monogenic signal with a band-pass filter and its applications to enhance weak and noisy anomalies. The Poisson scale-space monogenic signal with a band-pass filter is a 3D vector whose elements are the Poisson representation of the band-pass filtered data and its r x and r y components of the first order Riesz transform. In the wavenumber domain, we showed that these components are the multiplication of the Fourier transform of the filtered data by a Fourier-domain kernel, which in turn is the multiplication of the first-order horizontal derivative filter by the first-order vertical integral filter. This operation is stable making the components of the first-order Riesz transform quite insensitive to noise. The Poisson scale-space representation of the band-pass filtered data is given, in the wavenumber domain, by the difference between two upward continuations of the data at two at two continuation heights. By representing the Poisson scale-space monogenic signal with a band-pass filter in Cartesian coordinates, we can define three new filters: 1) the amplitude, 2) the orientation and 3) the phase. Tests on synthetic data showed the advantage of using the phase in the Poisson scale-space monogenic signal to enhance both strong and weak anomalies. In addition, if the data has not been correctly reduced to the north magnetic pole, the phase in the Poisson scale-space monogenic signal can be used to delineate geological structures produced by 2D sources like lineaments, faults and geological contacts. We applied the phase in the Poisson scale-space monogenic signal to real magnetic data from the Pará-Maranhão Basin in the Brazilian equatorial margin. Our test showed that the local phase in the monogenic scale space has a better performance than the tilt angle in enhancing the east–west lineaments produced by the magnetic expressions of the Saint Paul Fracture Zone cutting the continental margin.