I had a conversation recently with an executive about supporting characters (happily, this discussion came in the course of my starting on a pilot for them, after selling a pitch). An example he gave of a well-realized supporting character was Christian Slater's father in HEATHERS.
Now, it's been years since I watched the movie, but I remembered instantly who he was talking about. Don't you? Christian Slater's dad was single, rugged, damaged, and scary as fuck. You get exactly how his son got to be the way he is. I forget his specific role in the plot (Christian gets the explosives from him?), but I remember the shape of the character distinctly.
That's in a movie. Think how much more important regular cast and supporting characters are on a TV show, where you see these people week after week. Two of my favorite supporting characters on TV right now are on the same show, TRUE BLOOD. I like the main characters of Sookie and Bill well enough, but in the Southern Gothic by way of Twin Peaks world of Bon Temps, Lafayette and Amy stand out for me. They're both fully-fleshed characters, full of interesting, authentic character details.
Lafayette is gay, tough, a drug dealer and hustler who's tender and protective of his friends and can cook as well as he can lay asphalt for the parish. Also, he's funny and likable as hell. Jason's girlfriend Amy is a nightmare and I hate her, but she's a fantastic character. Earthy and hippy-dippy as only the entitled and overeducated can be, she's also completely merciless when it comes to vampires. She's a vegan and volunteers, but her compassion does not cover vamps, who she reasons are not alive and therefore undeserving of the same empathy you'd extend to, say, a trout. Amy's moral compass is absolute and she sees no contradiction in her behavior. And she's scary as fuck too.
It helps that Lafayette and Amy have things to do in TRUE BLOOD that are integral to the show's main stories; nothing's sadder than supporting characters getting relegated to B stories unstrung from a show's main plot or theme because the writers didn't know what to do with them.
Supporting characters need to do just what the name suggests: support the leads, the show, story, theme. And for that, they need three dimensions.