This is something I've never studied in any depth, I'm sorry to say. I can give you a basic skeleton, but few particulars.

Slaves have essentially no recourse to the courts. It's illegal to hear their testimony and as property, they have no standing to sue. It would be, legally speaking, like having your horse sue you. Even when slaves are suspected of doing something that threatens civilization as white southerners know it (chiefly conspiracy to revolt with a side-helping of killing all the whites) they're questioned informally and dealt with much the same. You might see an investigator or a panel look into things and hear what slaves have to say, but they're likely to use torture to make sure they have the "right" story and don't function as a proper court.

A person falsely held as a slave could file a freedom suit, though they would need a white lawyer to present the evidence. Southern courts do find for freedom on occasion. Solomon Northrup, who was kidnapped from the North and sold in Louisiana, prevailed and got this freedom after twelve years. Without a white ally willing to run the risk on his behalf, he'd have been a slave until he died or the Union army showed up. This is also true for Dred Scott, who had his former owner's family on his side.

The slave code generally applies to free people of color in the South. They're not slaves, but being legally black puts you under the code. They're not allowed to serve as witnesses, on juries, get an education, etc. A free person with the right friends and/or deep pockets could buy themselves partway out of those disabilities by retaining a white lawyer for representation and sue, but I can't say how they would have fared on a critical issue. Some free people own plantations of their own, so they clearly have a kind of legal standing that they can defend in principle against their white neighbors, but I've never studied the details. Those who prospered best usually, at least outside Louisiana, were blood relations to prominent white slaveholders that might act to bolster their rights.

Sources

Many Thousands Gone by Ira Berlin

Road to Disunion, Volume One by William Freehling