Some extracts from various responses I have written over the years::

......Many people who work in the singing teaching "industry" may be very practiced singers, but may not be trained at all in how to teach, -- quite a different skill indeed!
This is always an issue to consider when talking about any kind of teaching. 
We all know that someone may practice something very well, but it doesn't mean they can teach it!

......If you are looking for a teacher, you need to be very aware of what your needs are. For instance, some teachers are very interested in teaching a particular technique or approach to singing, but may not offer a lot of practical support around making backing tracks / rehearsal tapes to help you rehearse and practice the songs (if that is something you know you will need)

........I have often had students from other teachers coming to me to do some work on their voices, and I discover that even though they have been working with someone for some time, the student still does not have recordings of the songs to help them learn them. They don't actually know the basic notes of the songs they are trying to sing! (Of course, I'm speaking of students here who either don't have a piano or don't have the necessary music skills to learn the notes themselves.)

....Make sure that whoever you work with has really good piano skills, or has the facility to make good high quality recordings for you to take home to practice with -- They need to be customised to your key and style. Not just existing recordings that they have found somewhere - A lot of singing teachers who can't play the piano do this. They use karaoke CDs that they have bought elsewhere, all very nice unless they don't suit the key and the style you wish to sing in!

Do not waste time and money working someone who cannot do that for you
OR is unable to arrange for someone else to make recordings for you!.

This is essential, not just for you but for all students. It continues to horrify me that there are "singing teachers" out there supposedly "teaching singing" without providing any sort of recordings for people to take home. A pointless activity.

......You simply can't really learn to sing properly, if you're not even sure what notes you are supposed to be singing!

...........Some singing teachers out there, do not actually have any keyboard skills at all, except for perhaps the most rudimentary ones, and may not even be able to read music very well.  They may be fabulous singers, they may know a lot about singing, and they may be tremendous singing teachers as well, but they may not be able to offer you the support you need if you are not very skilled in those areas yourself.

..........If you do find a teacher that you like, and they don't have those keyboard skills, hopefully they will be aware of this, and will have an existing relationship with a pianist who will make backing tracks for their students.

If not, let me know and I can make recordings for you and send them to you via e-mail at competitive prices. (I may need to communicate with your teacher to get some information about key tempo etc) I frequently provide this sort of service for non-keyboard skilled singing teachers.  

I make the recordings for you and e-mail them to you as MP3s, and / or you can burn onto disk yourself. If the teacher tells me that you want to sing "Amazing Grace" in the key of G, I can make a suitable recording and send it across to you and you can use that recording as you work with that teacher.

......I'm always little concerned about any singing teacher who does not appear to have had a very wide ranging general music education, but I am willing to be convinced otherwise on that!!

......Someone may be a fantastic singer but it doesn't mean they're going to be a great teacher!  The world is full of fantastic singers who are absolutely hopeless at teaching! (although they may not know this about themselves, let alone admit it!) 

.......Avoid the teacher who appears to "demonstrate" all the time.
Meaning, most of the lesson seems to be taken up with them singing your song
AT YOU!  - "showing you how to do it" or else just singing along with you all the time.
(This can be a common problem with ALL music teaching, - especially the practitioners of more modern instruments, electric guitar etc, where many of the practitioners have no music education background at all!)
The goal of teaching singing is to get YOU to sing, and understand what YOU need to do to get your voice sounding fabulous, not just listen to someone else do it!

This is often a sign of the teacher who knows how to sing really well, but doesn't really know how to teach someone else how to do it!

Then again, be wary of the teacher who does not appear to be able to sing property themselves, (or appears reluctant to "show you how to do it") for obvious reasons! There are times when it will be important for a teacher to be able to demonstrate a particular 'tone' or technique to you, so that you can observe it in practice.

 ......Having spent a lifetime working first as a professional accompanist and trained school teacher, and later as a professional singer who has studied with many teachers, I have developed an approach where I can teach the technique of singing  clearly, and / or else I can provide what I call "vocal coaching" as well. 

......Vocal coaching is the simple teaching of the 'notes' and usually includes the making of recordings etc, to support a student in 'getting the notes right'. (This sort of work may be done by any experienced keyboard player; they don't necessarily need to know a lot about singing or vocal technique to do this)

......You need to work out what you need from a teacher.

......If you don't need support actually learning the songs, then you might get a lot out of working with the singing teacher who doesn't have wide-ranging musical skills, such as keyboard playing skills.

......You should feel quite free to experiment with a number of teachers before you make a decision to study with a particular person, including me!.
Work with a couple of people at the same time if you like, if it is not too confusing for you and their methods don't conflict too much.
 Have a few sample lessons with a number of teachers, and get a 'feel' for who you like the best.

......It is always best to shop around and make sure that you ask yourself questions about what's happening in the classes - what you’re happy with and what you're not happy with. Any teacher who resists you 'shopping around' should be avoided.  
While it is good idea to mainly commit to one teacher at once, any good confident teacher will not be threatened by you getting input / experience from other teachers. If they 'refuse to allow' you to do this, remind them that it is 'your buck' that is paying for all of this! -- and move on to someone else!!

....... Be very wary of anybody who asks you to pay large amounts of money upfront, paying by the term, etc. Do not do this, especially if they don't allow you to have a few "pay-as-you-go" lessons to see if the relationship is going to work for you.

..... Avoid shop-front "singing schools" - They tend to be expensive and are often staffed by a stable of unskilled young teachers as a moneymaking exercise.
Look for mature experienced professionals who offer personalised services.

......Never just accept everything at face value.  Always ask probing questions and always remember that it is YOU who is employing the teacher, not the other way around!

......REMEMBER who is really the 'boss' - YOU!

It's YOUR money you are spending, and you need to be satisfied with what you are getting.

......Some teachers make you feel as if you're ‘going to see the headmaster at school,’ and that you don't have the right to ask questions or question what it is they are doing and their processes. You won't get your money's worth if you don't participate fully in the process.

......A good confident teacher will not be threatened by you having a questioning ‘assessing’ sort of attitude towards them.

......Avoid people who appear to be overly bossy, or who give you the impression that they are doing you some sort of favour in teaching you at all.  There are quite a few people out there with a certain sort of arrogance (which often masks an inability to actually do the job of teaching well)  For the same reason, avoid people who insist that their way of teaching, or their method, is the 'best' or 'only' way of doing it.
Once again, making this sort of demand often masks a lack of confidence and actual knowledge in singing. 

......I have learnt that not everything works for everybody. 
......A good teacher will take the time to find what works for YOU. 
......A good teacher will not try to force you into a 'cookie cutter' mould
     of what they think you should sing like!

......Look out for the person who seems to be doing teaching with a sort of 'resentful'       mind set, based on the fact that they would rather be performing professionally       but will teach as a second-best alternative!!

......You need to be taught by someone who absolutely LOVES teaching and will 'LOVE" and CARE for YOU!