I came across one of the best nuggets of writing advice ages ago when listening to the Joss Whedon DVD commentary of a Firefly episode. He said that every character is the protagonist of their own story. As far as Jayne is concerned, everything happens to him first, then others.
Think about The Terminator. The main character is Sarah Connor. But from Ginger's POV (Ginger would be Sarah's best friend), this is her story. She's in a happy relationship with a decent bloke. She lives with her introverted friend who likes lizards. She loves her new Walkman and listens to it all day. After a nice night out with her boyfriend, she is suddenly attacked by a huge bloke wielding a Glock. He kills her boyfriend and shoots her in the back.
It's not the best story (which is why she isn't the protagonist) but it's a story. It has a constant state (happy relationship) and a trigger (the T-101 breaks into her house). She just dies very early on.
It's all about looking at the entire story from your sub-characters' POVs. The supporting cast are always tricky to get right. Some characters are there just for expositional reasons (to show that your hero only has one good friend). There's nothing wrong with that, but you run the risk of them being throw-aways.
So a good way to make them relevant is to check out what their story is. How would your script play out if John was the centre, not Jill?
What if Captain Dallas had survived in Alien? What would the story be from his POV?
Ron Weasley is the main dude in Harry Potter - what's it like being best friends with the most famous wizard ever?
What does Alfred do when Bruce Wayne is off saving Gotham City?
You don't need to go overboard with backstory or huge developments, but by thinking about sub-characters' story arcs, they will seem like far deeper personalities.
The last script I wrote has a female protagonist who is going down a particularly rough road in her life. That's where the best story is. I have two main secondary characters:
One is her best friend, who makes it his mission to protect our hero and drag her out of the life she's made for herself.
The other is a Priest who sees the bigger picture. He's not really too concerned with the people involved, but more on what their actions will result in. He has a past that is hinted at throughout and perhaps gets explained if you pay close attention.
But the point is - they both have stories. One goes on a mission to save his best friend, and the other sets out to save the world. All three stories (these two and my protagonist's) come to a head in the final act.
Make sure those minor characters are major in their own heads!
Over and out - shiny writing!