With examples from Saiyuki, iCarly, and Bones.
As, since and because all answer the question "why?" (Since can also be used to express time-- think of the expression "since 1937"-- but we won't address that here.)
Because is actually more common in spoken English than either since or as because it so clearly answers the "why" question.
"Why do we have to go to the Jeffersonian?" Parker asked.
"Because I need to pick up a case file," Booth explained, putting on the turn signal. "I'm not going to look at it while we're together, I promise."
"Can I look at the skeletons?"
Booth sighed. "Maybe."
As and since are often used when the reason is a less important part of the sentence. Garner's Modern American Usage advises that "In the causal sense, as should generally be avoided because (not as!) it may be misunderstood as having its more usual meaning 'while,' especially when it is placed anywhere but at the beginning of the sentence." It does work all right at the beginning of a sentence, though:
As he'd finished cleaning the dishes, Hakkai decided to take a walk.
"As" is also more common in British usage.
Since is more likely to have a clear consequence or result:
"Since I'm done cleaning the dishes," Hakkai said, "I'm going to take a walk."
There are some cases when since is clearer than as:
"We'll be taking a five-minute break," Principal Franklin said, "since Gibby needs to ... whatever it is Gibby needs to do."
"We'll be taking a five-minute break," Principal Franklin said, "as Gibby needs to ... whatever it is Gibby needs to do."
As in the second instance isn't as clear, because it could indicate the second usage of as-- "while."
The difference is pretty subtle and not always obvious. Generally, use what your instincts feel is right, or ask a friend or beta.