Hansey makes excellent points, and does it with fewer words than I do.
It depends how "supporting" they are. It also depends on how long the story is.
The latter first: In short stories, I usually limit my cast to three people. I figure I can do one person in 2000-3000 words, so maybe more for a little longer. In a novel, I might have characters who have a purpose, but do not play a major role.
If the character is part of the setting, I don't develop them too much. These characters can sometimes live as a "flat" person. The mailman who comes by five times in a novel, but does little else, can probably be described and left alone.
How to tell the where to develop a character? If the character feels flat and you feel it hinders the story, then they probably need background.
For characters who need more life, which is probably what you're asking about, I give them history. Again, depending on length, I give them the same treatment I give my main characters: I write out any information I would want to know if I met him/her/it/other in real life.
Birth place (town, hospital, room number if needed,) parent's names, family, places lived during life, favorite foods, hobbies, friend's names, memorable moments, girl/boyfriends, etc. I have never gone over 40,000 words doing this and I don't worry too much about the writing quality since it's only my reference material. 40,000 might sound like a lot for one character, but if he or she will be a supporting character at length through many stories or books, it can only help to know the character very well.
I might write a few lines of their typical/notable dialogue, just so I have a reference. This helps me separate the way characters talk, which I often struggle to differentiate. (I often get unexpected development doing this. Sometimes the characters say things I do not expect, but which reveal how they should act.)
I could write my real life friends into a story and, with some effort, create a realistic representation of how that person acts. Once I know the character, I can recreate him or her within a story.
If that isn't enough, I have a handful of scenes which I write a character into without the main cast. I won't necessarily use these scenes, but it helps me understand what the character would say or do.
Put a supporting person into a violent bar fight. How do they respond? Do they get killed? How about a warzone? A romantic outing? A boat ride with friends?
This may take a lot of time, so feel free to scale it down to the importance of the character.