"As the case may be" isn't common, but it definitely doesn't seem strange when someone says it. It is mostly used when there are two available options, and is used to stress the fact that a choice exists.
Imagine an event where some people walk, and other people run.
Someone might say "When you finish running or walking, as the case may be, you can come inside and drink hot chocolate."
Here, "as the case may be" is just used to show that two options exist.
Often, we use it after we FORGET one option, and want to correct ourselves.
"When you finish running---oh! or walking, as the case may be--you can come inside and drink hot chocolate".
It is quite often used humorously, to show that you think the second option is strange or bad, but you are pretending to treat it equally. Imagine that Bob lives in his car instead of a house, and you want to make him feel ashamed (of course, you would never never do this, but some people are evil)
You could speak to a group, and say,
"After you go back home to your house--or your car, as the case may be--you should get some sleep."