I've had a total of 3 with AA direct booked flights in the last decade
Most recent two appeared on the AA app closer to the departure date rather than say 2 days later like BA might do. First one was offered at a self-check in kiosk.

First years ago from DCA to somewhere; second from RIC to BWI or somewhere last year ($50 or so, but got upgraded anyway); third one was a longer domestic, maybe a transcon (flat bed may have have decent enough for $250?, but I can survive a Y seat).

Didn't take any as there's no additional tier points for BAEC members. You only get, say, the 10TP for the economy fare rather than 40tP for the first class cabin upgrade.