There are two – everyone else is not them but want to be:
Berklee College of Music in Boston for commercial music
The Juilliard School in New York for classical tradition

There are some conservatories of music that are also good but don't have the same weight as those two:
Oberlin, Peabody, Curtis, Shenandoah, Cleveland, Boston, Eastman, Manhattan, Thornton, etc…

In California:
Thornton at USC, UC-Berkeley, SFSU

"whatever else it is musicians take…" would typically be:

4-6 semesters of music theory and applied musicianship, 8 semesters of applied music (lessons), 8 semesters of ensembles, 1-2 semesters of conducting, 1-3 semesters of advanced theory/arranging/composition, 1-2 semester of class voice or class piano (or both), 2 semesters of music history, 1-2 semesters of music literature, 1 semester each junior and senior recital, …

Pedagogy courses for music ed types, additional applied and ensembles for performance types, more composition and theory for comp/theory types, more history for the musicology types, and churchy stuff for the church music types.

Understand though – almost every college has a music department that teaches theory. You don't have to be a music major to take those but you will need permission from the department.

Check with the head of the Music Department at any local college or university that offers music as a degree. I have found them to be great resource in find voice teachers and vocal coaches.

If they offer a Vocal Pedagogy course, (this teaches voice teachers how to teach voice) you might find you can volunteer to be a students student, and even get free lessons. (This is how I got started with voice lessons.) But at the very least, the department usually has a good bead on the better teachers in the area.

Check around with the colleges and universities in your area, and find which ones offer degrees in "Vocal Pedagogy." This is a music degree program that teaches students how to teach voice.

I got two semesters of free lessons by volunteering to be a pedagogy major's student. Each student in this degree plan must have a student to work with. I got into it when a voice coach for my chorus made the same suggestion I am giving you now. I went to the University of Colorado's (10 miles from where I live) web site, and found that they offered Vocal Pedagogy. I checked out the degree plan and found the email address of the head of the department, and sent her an email offering to to be a student's student if someone in her program needed to find one. The only thing I had to agree to was to show up for class two times during the semester so my teacher could demonstrate how he was teaching to the class.

I spent two semesters as his student and didn't cost me a penny. After he graduated, I continued on with him, paying for my lessons. Worked great. Anything I needed, if he did not have the skill or knowledge to fix, he would take it to his teacher. His teacher even came to our lesson a couple of times to help work on my falsetto.

For recommendations for a vocal teacher, check with the choir director at your local High School. They are always plugged into the teachers in the area. Also as mentioned above, any college or university that has a music degree, will also be well plugged into the best teachers in the area.