When looking for posts about good running form, you will end up finding a lot about where on your foot you should supposedly land, what movement your legs should do, and sometimes what you should do with your arms. But surprisingly, there is very little out there about what your spine should do. In fact, there are many articles that imply that any movement at the spine would be inefficient and that you should aim to fight any bends or rotations and pretty much move like an old school walking robots.
Many cross-training programs for running also seem to be designed to train that kind of spinal immobility: planks, anti-rotational exercises, and two-legged weight lifts with no movements at the spine are very common.
In a way, if we think of running as moving the legs to get forward, it kind of make sense to want to isolate the movement as possible and not "waste" energy in moving muscles that do not get you forward. But that idea comes from a misunderstanding of the movement of running. Despite all the sideway view images and videos of runners making it appear as such, running is not a 2D movement forward: we are not cars on wheels. We need to shift our weight from our left foot to our right foot and rotations actually play a big part in us moving forward.
Now, there is also such a thing as too much movement at the spine, and in that case, focusing on stability makes sense, but with the exception of the anterior pelvic tilt, I have found that most runners are more likely to be too stiff than too mobile. Next time you go for a brisk walk or run, pay attending to your torso. Does it move or does it just stay square to the direction where you are going? Now look at the image below, this is how your torso and shoulders should move in relation to your pelvis when walking or running: