Qualifications
There are two primary roads to earning the title of aviation safety inspector: flying and fixing. Before anyone is going to be comfortable with you telling pilots and mechanics how to do their job, you're going to need to prove that you could do those things yourself. And the best way to prove that is to, well, do them yourself.
Unlike learning to fix a car, learning to fix a plane can often require a more formalized approach—after all, most of us don't have a 242,000-pound 787 Dreamliner in the front yard to tinker with after school. You can get a good head start on things by pursuing an aviation maintenance or avionics degree through a technical college, or by weaseling your way there through carefully considered apprenticeships.
Pilots have it about the same. They'll need to get a pilot's license, find a job that grants them at least a year in the air piloting a massive plane, and bank at least 15,000 flight hours (source).
And for both of these paths, that's just the start.
The FAA is a regulating body, so it should come as no surprise that if you want to work for them, you'd best be prepared to be regulated yourself. That means certifications. First, you'll need to earn a certificate from an FAA-accredited program. Which one you need depends on the specific job you're looking to perform. Generally speaking, though, you can expect a year or two of classes with a minimum of 1,900 hours in coursework (source).
And we're still not done.
Not only do you need the work experience, and the hours, and the certifications, you'll also need to hold said certifications for three years before they'll even consider you. But don't fret, there's good news too. Once you're done, there's a handy online application form to fill out on the FAA's site, so you don't have to bother printing a . Isn't that convenient? It's going to save you, like, up to fifteen minutes.