REAL, PROPER and PROFESSIONAL Singing lessons should teach you the proper way to breathe and pitch from the diaphragm - NOT FROM THE NOSE OR THROAT, NOR BY SHOUTING! You should also be taught the rudiments of music, and not just singing techniques. Your teacher should be knowledgeable enough, and should have experience in singing in different genres, to be able to teach you to sing in the appropriate way for the style of music you like (be it pop, classical, jazz, R&B, etc.) In this way, you sing the right way for the genre, and you learn to project your voice, learn techniques to make your voice sound more solid, enhance your tone, extend your range, pitch accurately, express yourself beautifully and you also really learn to read music like real musicians do.
SINGING LESSONS ARE NOT JUST ABOUT SINGING ONLY!
The human voice is an instrument, and whatever you are singing, there are corresponding notes to it. You should learn the notes to what you are singing!! If you sing without learning the notes to what you are singing, then you are just like a parrot, mimicking sounds! So, in addition to proper breathing and pitching techniques, you should be taught the rudiments of music (using either the piano/keyboard/guitar) - for you to learn to read notes, to know the keys to your songs, to sightsing, to count and differentiate time signatures, learn how to sing and hear harmony, etc.
You should be taught the techniques well enough and gain a very solid foundation, so much so that you will be able to sing in any genre - whether pop, classical, jazz, etc. It is only if you decide to concentrate on one genre that you love very much eventually, only then you can concentrate and specialise, and your teacher will support you. Good technique is good technique. And good technique enables you to sing anything and be versatile!
FUNDAMENTALS OF A GOOD SINGING TEACHER
If your teacher has no knowledge of reading music, and cannot accompany you on piano/keyboard/guitar, and is grabbing sounds from the air to teach you, this is an inadequately trained singing teacher! All the techniques and rudiments should be taught with the teacher playing the piano/keyboard/guitar for you. She/He should teach the techniques using scales, while checking your pitch against the piano/keyboard/guitar.
How will your teacher and you be able to tell if you are in pitch, unless you pitch it against an instrument? Your teacher should also be able to accompany you while you are singing. Teachers who teach singing using only instrumental tracks and are not able read and play music themselves, are not able to read the charts to your songs and and are not able to accompany you, are inadequately trained to teach singing.
Your teacher should be able to tell and teach you in what key you are singing. She should be able to tranpose keys when accompanying you, to bring the song down to a key comfortable for you, and she should be able to teach you to sight-sing.
CLASSICAL SINGING IS DIFFERENT FROM POP SINGING!
Classical singing techniques and tones are different from pop and jazz technique and tones! Although scale training exercises and breathing exercises are similar for classical and pop/jazz singing, your teacher cannot be training you in a classical way only, then proceed to let you sing pop songs using classical techniques only! Your teacher should not let you do pop songs with your head or choir voice only. These are used mainly in classical music! Vice versa, neither should she teach you to sing with your chest voice only. Pop and jazz and R&B call more for a "conversational" style of singing, with some head voice thrown in, and that calls for special belting techniques THAT DO NOT MAKE YOU SHOUT. Your teacher should be able to show you the similarities and differences between classical and pop/jazz singing,
If your teacher is not able to correct you if you are shouting, or if your throat gets strained and sore while you are singing, then you are not being taught the correct techniques. Like teaching classical piano is different from jazz piano, your teacher should be able to differentiate and demonstrate the differences between the various genres.