There are almost as many ideas about correct breathing for singing as there are voice teachers. Every vocal pedagogy book has a chapter about breathing, and the various approaches range from “Don’t think about breathing at all,” to the most complicated and intricate understanding of the musculature involved and how to manipulate the muscles for proper breath support. The latter approach focuses on the use of intercostal muscles, expansion of the rib cage, use of the pelvic muscles, and raising the chest. Although none of those things is necessarily wrong (indeed, some of those ideas are provably correct by scientific measurement), they are simply descriptions of what happens when we sing correctly, not what we do to sing right.
The more simply we think about breathing, the better we will do it. However, correct breathing is not so simplistic that we don’t think about it at all. To have the kind of power necessary for singing, the voice must speak clearly, which creates an intense muscular vibration. To balance and buoy that intensity, the breath must flow freely and consistently, which requires intense engagement with the breathing process. While singing, we cannot forget about breathing altogether, but we must not focus too much on the physiology of breathing.
Smith, W. Stephen. The naked voice: A wholistic approach to singing. Oxford University Press, 2007.