Alot of exercises that arent going to make sense to you but trust me they are important. They will also teach you proper warm up exercises and those are key to singing.

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Going by the exercises I've done for acting, the main exercise is the controlled breathing. It's important that you breathe from the stomach, that is your stomach rises when you breathe and not your chest. Other than that, I have no doubt the exercises will vary depending on the instructor.
Vocal warm up exercises for a start. Then, you'll probably move on to breath control and techniques on how to maintain your breath. Then, you will have to practice along with a musical instrument (this is to "tune" your voice to the proper key). Then, you'll probably get a list of words to sing over and over again. Next time you come to class, you'll be expected to sing these words to the rest of the class.

I took singing lessons about 3 years ago. They helped quite a bit.