Probably, you would look for the same things you would seek in any private music instructor. No teacher is right for every student. Here are a few things I would suggest:
- Ask around. Do you know anyone who has been a voice student? Is there any particular instructor that this student recommends?
- When you do have one or more instructors in mind, find out their background and experience. Do they have experience in solo performance? Choral performance? Opera? Music theatre? What is their level of education? None of these answers will be determinative by itself, but finding a teacher who has been where you want to go can be a good thing.
- Is it going to work for you to have a lesson with this instructor basically every week? There can be a variety of reasons this is not the case. The teacher could be too far away from you for you to be able to take the time and expense to go to the teacher's studio each week. Or, the teacher could have too much else going on and could constantly need to reschedule or miss weeks. Everyone needs to reschedule once in a while, but doing so constantly is a problem, as it breaks continuity that you need for optimal learning.
- Is the teacher organized? Are payment and other policies clear? While the business acumen of a teacher may not directly correlate to his or her teaching abilities, a decided lack in this area can lead to a lot of frustration on your part. Also, there is a certain effect that this structure has on a student/teacher "separation" that can be beneficial.
- Can you afford the teacher's rate? The cheapest teacher may certainly not be the best teacher, but if you cannot afford lessons consistently, you will struggle.
- Does the teacher have references? Not all will, particularly if they are new to setting up a studio, but it is something to keep in mind.
- Once you have a list of instructors narrowed down a bit, see if you can try a lesson or two. Don't be afraid to pay for the teacher's time; back when I taught a lot more private lessons, I did offer a "first lesson free" deal to pretty much anyone who asked (or saw the offer in an ad). However, not all teachers will want to do that for a variety of reasons, notably that doing so tends to encourage students who are not serious about lessons for one reason or another. If the lessons "feel" right to you, and you know that you are learning something (and what you are learning), this might be the right teacher for you. If you have difficulty understanding the teacher, or the teacher cannot answer your questions, this might not be the right teacher for you.
- The teacher should be willing to discuss your goals and learning style, and the teacher should be able to articulate their own methodologies and reasoning behind them (at least upon request).
There are a plethora of items to consider, but the main idea is to find a teacher that works for you, with whom you are comfortable, who teaches in an environment comfortable for you, and from whom you find you are learning.