You can file a lawsuit of course. The question is whether you have any evidence whatsoever that they didn't hire you because of illegal discrimination. As you say yourself, they didn't hire you "for no apparent reason". It is entirely legal not to hire you "for no apparent reason". Without evidence of illegal discrimination that court case will go nowhere.
Consider that there are jobs where 100 people apply, and a dozen people are interviewed. Getting not hired three times, when 12 people got interviewed, is just normal. If the same 12 people were interviewed at each job, then you and eight others were rejected each time, because there were 12 good enough to be interviewed, but only one job. Not being hired three times is nothing. Maybe instead of thinking of lawsuits, you might think of some training to do better in interviews.
So not only do you not have any evidence of illegal discrimination, it's not even reasonably likely that there was illegal discrimination. Are you really claiming that all the three times where you didn't get the job you were the best possible applicant that was there? Did you check who got the job? Maybe it was a gay black jewish woman in a wheelchair who was just exceedingly good at her job?
The point is really: Belonging to some protected class doesn't give you a right to get the job. It gives you a right not to be rejected because you belong to that protected class. It doesn't give you a right not to be rejected for any other reason. And being rejected without being given a reason is no proof, not even an indication, of some illegal discrimination going on. The most obvious and common reason for rejection is that out of all the applicants, you were not the best.