You are correct that hardly any case law exists built on the 3rd Amendment. It just doesn't come up very often. However, there are a couple cases where it has come up.

The famous case Griswold v. Connecticut, 381 U.S. 479 (1965), created a "right to marital privacy," which invalidated a Connecticut law banning contraception. Justice Douglas wrote the majority opinion and cited the Third Amendment as one of the specific provisions that, taken with others (like the right of association under the First Amendment, the Fourth Amendment's rights against unreasonable searches and seizures, and the Fifth Amendment's right against self-incrimination), created a right to privacy under the Constitution. He spoke of "penumbras, formed by emanations from those guarantees [in the Bill of Rights] that help give them life and substance." Id. at 484. Importantly, the right to privacy recognized in Griswold was expanded to eventually become the basis of the right to abortion in Roe v. Wade, 410 U.S. 113 (1973).

As consequential as Griswold is, it's not really a 3rd Amendment case. It's really just a case where a Supreme Court justice mentions it as a small part of a much larger argument. Perhaps the only case that actually turns on the 3rd Amendment is Engblom v. Carey, 677 F.2d 957 (2d. Cir. 1982). In the case, some corrections officers at a New York state prison were on strike. They lived in apartments in a building on the prison facility, but as a result of the strike were prevented from entering their apartments. The state called in the National Guard to run the prison during the strike, and decided to house the National Guard in those same apartments. The corrections officers brought a suit under the 3rd Amendment.

The Second Circuit (note, NOT the Supreme Court, this is an intermediate Court of Appeals whose jurisdiction only extends to the states in its circuit, here Connecticut, New York, and Vermont) ruled first that National Guardsmen are indeed "soldiers" under the 3rd Amendment. Then it ruled that the 3rd Amendment is "incorporated" against the States (a fancy lawyer way of saying that it applies to the States too, not just the federal government. Note that the Bill of Rights did not initially apply to the States, it only applied to the federal government, which only changed right by right over the course of the 20th century). Then it ruled that the 3rd Amendment's protections apply not just to owners, but to those who can claim "lawful occupation or possession with a legal right to exclude others." Id. at 962. Basically, it applies to renters, not just "owners" as the bare text of the 3rd Amendment states.

This remains one of the only cases to ever turn on the 3rd Amendment. Technically speaking its rulings are only binding in the Second Circuit, not the entire country, as the Supreme Court did not take up the case and has yet to consider another 3rd Amendment case.