There is almost no jurisprudence surrounding the 3rd Amendment. The most famous case (because there are practically none) is Engblom v. Carey, 677 F.2d 957 (2d Cir. 1982). The facts in that case revolved around a prison strike. Two guards lived in employee billets on the premises of the prison. It was an option available to guards but not required as a condition of their employment. They were treated as tenants and had $36 deducted from their paychecks for the housing. It was essentially a landlord-tenant relationship and the $36 represented rent payments.

There was a statewide strike of guards and eventually Gov. Carey called in the National Guard to staff the prisons, including the one at issue. During this time, the residents of the on-site housing were denied access and the National Guard was placed within their units.

The Second Circuit noted in its opinion that there was an "absence of any case law directly construing this provision presents a serious interpretive problem, and little illumination can be gleaned from the debates of the Constitutional Convention." Like I said, this is not a commonly brought claim.

The court examined whether the 3rd Amendment is meant to only apply to "Owners", that is, fee simple owners of property (do you own a house). They concluded that was not the case, and tenants were entitled to a right to privacy and protection against the quartering of soldiers (which they determined the National Guard qualified as). However, it's important to understand that the appeal came as a result of a district court ruling on a summary judgment motion, which the district court granted in favor of the defendants. Based on the facts, the court concluded that "we cannot agree with the district court's finding that appellants' occupancy was more analogous to a possession incident to employment."

They remanded the case back to the district court where the 3rd Amendment issue was dodged by concluding that the state actors had qualified immunity because they did not knowingly act illegally and they could not have knowingly acted illegally...because there was no precedent or case law surrounding the 3rd Amendment. Essentially, it was a novel situation and they did not want to punish the government for acting in a legal vacuum.

But, it's important to note that the Supreme Court has never issued an opinion on a 3rd Amendment claim and this is virtually the only case law surrounding it. You can read the text of the opinion here