Wow. I say wow because I saw this question as I was reading about this in a history book. The book, by the way, is a textbook I have for my Antebellum (Pre-Civil War) America course. Called "Becoming America" by Henkin.
Anyways, on August 22nd, 1831, Baptist preacher-slave Nat Turner (believing he saw signs in the skies) killed his master. He had planned this "uprising" from February reportedly, and he gathered 60 slaves or so and managed to kill 50+ white men, women, and children. Turner escaped, despite vigilante groups trying to track him down. It took them two months to find him, and he was eventually killed. Apparently, up until then, not a single white person had been killed in Virginia during the entire colonial era by a slave revolt, a period of 170 years.
I haven't read enough to talk about more, but that's one tale at least that demonstrates that eventually, slaves did revolt. I can recall more, but I would hesitate to recount hazy details until I can check sources.
Edit: Thanks to /u/Irishfafnir for pointing out that I'd misread the text and forgotten to include the "in Virginia" portion!
I am not sure of the regard in which he is held, but Herbert Aptheker documented over 200 slave rebellions (American Negro Slave Revolts is the book).
Additionally, Daniel Rassmussen argues in American Uprising that an 1811 slave riot was really an organized rebellion.
So yes, there were definitely instances of slaves revolting in the antebellum South. The Nat Turner rebellion, mentioned above, is probably the most famous. I am unaware of any being successful.
The fear of rebellions was something on the mind of white Southerners, and they passed laws and developed systems to prevent them. The laws preventing teaching slaves to read were enacted in response to this as well as laws limiting slave assemblies.
I'm sure that others can expand further, but as a very rough first pass there is a set of articles on Wikipedia covering North American slave revolts.
Edit: I realized that there may be a successful revolt that I didn't think about - The Amistad. That's very limited though, and hardly a "Spartacus" moment, as it didn't spread.
While not in the United States, Toussaint L’Ouverture is frequently referred to as the "black Spartacus." He organized and led a slave army which fought French control of Haiti in beginning in the early 1790s.
After defeating the French and bringing abolition to the island, his army then faced British invaders in 1798, who were expecting to find a demoralized French force and instead found a highly organized army of 55,000 slaves. Keep in mind that George Washington never commanded more than 20,000 troops.
He defeated the British in a series of seven battles, forcing them from the island. Two years later, he faced a Spanish invasion, and again successfully repelled a major European power. In 1801, Haiti declared independence.
In 1804, Napoleon returned to power and sent a huge army to re-take the island. Toussaint sailed to a French consular ship to negotiate, but he was arrested instead. He died in a jail cell high in the French Alps.
I would definitely check out this well-sourced article which gives the details behind this amazing story.
Denmark Vesey. He planned a rebellion to kill white people in Charleston, South Carolina. Unfortunately he was caught and executed before the plot could proceed. There's a statue of him in Charleston, and it's controversial to some white people living there.
http://www.britannica.com/EBchecked/topic/626831/Denmark-Vesey