The airlines will have access to several possible sources of info...

State/local records. This will be based on where you report that you have lived. All airlines may not bother with this...in sme cases they can look up public records online but in many places you have to physically visit a courthouse to pull state or local records. Majors are more likely to do this than regionals.

Background check services. These may collect and store copies of court records from numerous locations and match your name for a fee. Not all locations allow their records to be downloaded, but you can google these companies and for a fee run a check on yourself to se what comes up.

FBI Database. Airlines get to see this because of the TSA background check requirements. However...even if you meet TSA standards the airline still sees your record. In the case of some deleted records, the record will still exist but no details will be visible. An airline will only know that "something" happened on that date...and will wonder why you didn't tell them. But most misdemeanors do not appear in the feds database anyway. You can get a copy of your own record.

But with all that said, you are really going out on a limb if you lie about your background. For one thing, that record is probably reportable on a FAA medical exam (read the fine print). If you lie on that form, your misdemeanor just became a felony (the FAA arranges to throw people in jail for that).

I assume this incident ended your military career? Is there any documentation, such as a non-standard separation code on your D-214 or a general discharge?

How old were you when this happened? If you were under 25, an airline might just write it off to youthful exuberance. But over thirty, they will assume you have issue(s). It's one thing to get drunk, a lot of us do that now and then, but in order to get a PI you usually have to tick off a cop.