You can check AA availability on AA.com. You can check BA availability on BA.com, but you have to get a BA FF# to do so (just apply on the site is the only requirement). I have not tried to get IB info, but I'm sure there are ways. Expertflyer may also have availability info. I prefer to choose my carrier/flight and get my itinerary all laid out before calling, then just read off the flight numbers and dates and have the AAgent book it for me. YMMV.