Well, as always the answer is: it depends!
For example, if you want to compute a single interferogram after an Earthquake or volcanic eruption, where the displacement is much higher than the normal atmospheric phase screen, you could avoid to do any atmospheric correction and consider that you are only measuring deformation (but there your assumption is that deformation>> atmosphere )
For SBAS, they normally estimate the APS of each single image by mathematical approaches, and they are normally well calculated, when the number of images where each single image is involved is relatively high (> 6/7 ifgs).
For PSI the only APS which can be well estimated is the one correspondent to the master image, and hence all APS for the slaves will be on your results. You should be at least aware of that. You can always remove it or not (depends on specific application) but you should know that the slaves APS is still there. Methods are many, you can just look around in literature. My suggestion is to use TRAIN from Dr. D.Bekaert, a very good friend that did a very nice work for his PhD creating this Tool for Reducing Atmosphere InSAR Noise (https://github.com/dbekaert/TRAIN) that is nicely working with StaMPS.
Good luck!