It seems to me that the different viewpoints are all about the amount of distance between the reader and the characters. A story written in first person has the least distance because the reader is right there in the character's head, hearing his thoughts, sharing his emotions. It's very different from the omniscient story, where there is a narrator standing between the reader and the characters; in these stories, the reader is often outside the viewpoint character, watching what's happening instead of experiencing it along with him. Third person is obviously between the two: more intimate than omniscient but more distant than first-person. So, when choosing which viewpoint is best for your story, it all depends on how close you want your readers to be to the characters.
A great example of YA written from the first-person viewpoint is Speak, by Laurie Halse Anderson. The story is about a girl who's tormented by a secret, and this viewpoint allows the reader to slowly learn what the secret is while becoming more and more empathetic with the main character along the way. A different viewpoint would not have let us into her thoughts and emotions the way the first-person does; it also would've made it harder for us to figure out what exactly had happened to her, since she refused to talk to anyone about it. So this viewpoint is perfect for when you want your readers to climb right into the character's skull and roll around in her thoughts and experiences.
No clue. Seriously. Except it's maybe good for short, short, really short pieces.
On the other hand, there are times when you want to maintain distance between the reader and the character—if your character's a psychopath, for instance, or a liar and therefore unreliable. In Tad Williams' Otherworld series, one of his viewpoint characters is a serial killer named Dread. He's a well-rounded, interesting villain, but a sick twist, just the same. Personally, I wouldn't want to be inside his head; I appreciate the author's use of third-person to safeguard me from that. Another reason to go with third-person is if you want to tell the story from a number of different characters' viewpoints and you feel that first-person would be too intimate for jumping back and forth. The Sisterhood of the Traveling Pants books are examples of how stories can be effectively told from the third-person viewpoints of several characters.
In my opinion, omniscient is best used when a) you want to create a narrative feel or b) you need to impart information that's beyond your character's experience. The Hobbit is a good example. There's so much information and history that Bilbo, in his happy hobbit world, couldn't possibly know; the way Tolkien chose to write the story, an all-knowing narrator was necessary to get the info across. A more recent book written in omniscient is DiCamillo's The Miraculous Journey of Edward Tulane. It has the feel of a fairy tale or long-long-ago type story, so this viewpoint works well. And because the main character is a stuffed rabbit with limited knowledge of the world, writing in the omniscient allows the author to reveal things that are outside of Edward's experience.
When choosing a viewpoint for your story, keep in mind the specific challenges of each. In first person, you can only share information that is known by that character, which can be constricting when you're trying to write; the voice also has to be spot-on, since you're not just writing about that person—you're writing from inside his/her head. Third person can be tricky because once you decide which end of the limited/omniscient scale you want to employ, you've got to stick with it. If you choose a limited viewpoint, you can't show anything that's outside the experience and knowledge of the character you're writing at that time. Omniscient may seem like the easiest to do—kind of an anything-goes style of writing—but you have to work hard not to bog the reader down in backstory and long, drawn-out passages of narration. It's not easy to write omniscient in a way that keeps the reader engaged and connected to the characters.
So when choosing which point-of-view to use for your story, keep in mind the amount of distance you want to establish between your characters and the reader. Think about what kind of story you're writing; if your main character is killed by radical heartburn in the second-to-last chapter and won't be able to finish telling the story, first-person probably isn't the way to go. And let's be honest: if your main character dies by something as innocuous as indigestion, you may want to rethink more than the point of view.