You should make progress at every single lesson. However, coming in regularly is super important to make the most of your investment. For most people, that’s every week. Some people like to come in twice weekly (but you have to make sure you have time to practice between lessons) and some people come every other week (but you have to make sure you stay on track with your practicing and don’t just practice once every 2 weeks the night before your lesson).
Professional singers often come in just “once in a while” but for beginners and it’s better to have consistency for building your skills over time.
How many lessons you need is going to kinda depend on your goals. You should notice a change in the first lesson, though. You should notice pretty significant improvements by the 10th lesson. However, other things to watch for are improvements of how easy it is to sing. Often clients will come in having a hard time getting their voice to leave their throat and come out into the room. In addition to how you sound, improvements in that are one of the hallmarks of vocal lessons.