Breathing = Singing.
Most, if not all, of my beginner students start off their first lesson trying to squeeze high or long notes out of their inexperienced throats after taking either no breath at all, or merely breathing like they would if they were having an off-the-cuff conversation with a friend. I think this problem stems from not really realizing that singing is completely different from regular speech and that it demands a unique manipulation of the vocal chords, respiratory system, mouth, and brain.
Some students are suprised to receive a biology lesson from me at our first meeting, as I have been known to occasionally go into graphic detail in describing the scientific reaction your vocal chords go through in response to improper versus proper singing technique. But when the creepy part is over, the student comes away from their first lesson well informed of the dangers of abusing your voice. Towards the end of class, with a simple demonstration through scales, I show them the before-and-after effect that healthy singing practices has on the sound quality, power, and range of their singing voice. I say my next piece with literally no exaggeration whatsoever, every single solitary student I have ever taught sees a noticeable result within the first class after learning this technique. Usually this improvement is a significant increase in range, sometimes upwards of half an octave beyond their raw, un-coached range. Small imperfections in technique are quickly and easily eradicated with this technique, as well as noticeable increases in volume and power, and pitch accuracy.
It may seem far fetched that one simple rule can fix so many issues, and while it won't fix every single issue, it's the strongest start you can get towards a consistently beautiful sounding singing voice. Another wisdom I have learned throughout my studies which holds significance on this rule: A simple design is often the most divine.