The correct breathing coordination for singing is absolutely essential in order to produce a free, ringing, beautiful sound. I had already explained in my other post on the breathing what the correct set up of those muscles is and how to do it properly. Now I would like to discuss one of the most destructive ideas about the breathing that I sometimes hear being taught to students; i.e., pulling in the stomach when singing.
Pulling in the abdomen when singing causes there to be a collapse of the inpiratory tension. As a consequence of the inspiratory (abdominal expansion) collapse, the singer is forced to lock the throat causing a valvular action - the Valsalva maneuver - whereby the glottis is closed too much and the singer is constricted. The singer is driving the air out against a closed throat. This happens when someone grunts, lifts something heavy, coughs etc. And you will notice when you cough that the stomach pulls in. It does this naturally so that the diaphragm is jolted by the abdominal muscles, thereby propelling the air out. We then momentarily close the glottis, building up pressure, only to release it explosively in the cough. This is how our bodies attempt to get something out of the throat.
Now, if we were to hold the abdomen out - engaging the inspratory muscles - while trying to cough we will notice that it is nearly impossible to initiate the action. This is due to the fact that the holding out does not allow the abdominal muscles to pull in too aggressively against the diaphragm - which is what is needed in a cough.
In singing we do not want the throat to close. That would only limit the vocal folds from vibrating freely. Instead, we want the vocal folds to vibrate completely unencumbered. So it is imperative to keep the abdomen expanding during singing so as to prevent the throat from closing. Thus, the holding out of the inspiratory tension (abdomen expansion) allows us to hold the throat open and vibrate the vocal folds freely.
Additionally, the balance of the inspiratory tension against the expiratory tension affects the vibrato action. Most of the singers who pull in the stomach when singing either already have a tremolo problem or soon develop it. That is due to the fact that the balance of the two tensions affects the vibrato action directly. There is a pulsing again that happens when you hold the inspiratory tension against the expiratory tension which directly affects the amplitude of the vibrato. And this is especially apparent on high notes or loud notes. Therefore, collapsing the expansion of the abdomen does not allow for this to happen naturally, how it should, but rather will cause the singer to develop a tremolo over time.
In closing, please never listen to any teacher who tells you to pull in the stomach or abdomen when singing. It will only cause constriction and/or tremolo issues with the vibrato. Proper appoggiare is the abdomen holding out.