I've flown all three and I'd say that BA is the hands-down best of them. The seats (AA suites, not coffins) are comparable on all, with perhaps the slight edge going to AA, but in most other respects, I find BA superior. The catering, ground handling, lounges and service consistency are all stronger on BA. Although I was very pleasantly surprised by the excellent catering on a recent UA flight, the food is generally more interesting on BA and the wine list light years ahead of the US carriers. Another deciding factor for me is the absence of curtains between the cabins and prohibition of walking around the plane on US carriers.

Have to admit, I have not flown AA to/from LAX, and I gather LAX has one of the best FLs on the network. The BA F lounge at LAX - like most there - is really nothing to shout about.