Why phase information is usually ignored after Fourier transform?
Simply put:


Because in interesting cases, it remains difficult to interpret, and people tends to use other tools
Yes, potentially a lot of non-stationary features, timings, onsets, discontinuities


For a basic use, like measuring the relative energy of different frequency component, the phase is of little use. As waves tend to last over several samples, sliding/overlapping versions can help retrieving timing/localization of features. 

However, phase contains a lot on information, especially on images, and perhaps plays an important  role on perception (e.g. The importance of phase information for recognizing natural images). The importance of phase is recognized in many works, see for instance The Importance of Phase in Image Processing, and it is still an active topic. Multiscale phase might detect onsets, contours, while the distribution of phase coefficients could be used for texture retrieval. The phase computed is wrapped  between $-\pi $ and $\pi $, and unwrapping it in 2D or 3D seems to remain challenging. Hence, some resort to complex transformations that are more localized than Fourier bases, like filter banks or "wavelets", to capture phase information at the correct scales.