In White Wolf's games set in the World of Darkness, it's customary for each player to run through a series of vignettes from their character's background with the Storyteller. The kinds of things they focus on might be different for everybody - if one player talked a lot about his character's family and how close they were before he became a vampire, maybe that's worth digging a little deeper into to see if maybe the supporting cast of spouse and children - who may only currently consist of names and ages jotted down after a moment's consideration - have more going on beneath the surface.
Or, if the character has a lot of dots in Knowledge skills but only tangentially referenced his college years, fleshing out that span of time could be constructive. It may be that a character's first brush with the supernatural, no matter what kind of game you're playing, was not the catalyst for that person's transformation into a PC. For vampires, for example, the being that kills someone and brings them back to "life" (known as their "Sire") is a really big deal. It may be that, although the character didn't piece it all together until after his or her death, there were multiple encounters with his or her Sire from early childhood onward. This can help develop who is ultimately the most important person a vampire knows after returning to the world as one of the unliving into a really three dimensional supporting NPC instead of just some plot vehicle for a life changing event that will never come up again.
In my games I like to do a little back and forth with players before they join the campaign, probing with questions about this or that aspect of their personality or history and seeing how it ties into the crunchy bits on their sheet. If someone picks the Poverty-Stricken trait, I'm curious to know more about that and whether that was the sum of their experience before the game started or if a reversal of fortunes occurred that saw their family's wealth and influence precipitously decline. There might be more to the story or maybe not - maybe the character doesn't think too much about where he came from, but even that can tell me something important about him.
As a player, how much do you welcome the DM to get down into the mud with you when going over your character sheet before play? Do you like stuff like this or do you just want to hit the ground running?