Flights on AA, its oneWorld partners (note: not all its partners) and Alaska Airlines count toward status, including the Challenges.

One exception that may affect you is BA flights to/from the U.S., which earn no AA miles for reasons discussed at great length in other threads. This can be avoided by either flying AA across the pond or flying BA to Canada or another North American destination.

Also, some deep-discount partner economy fares earn fractional, occasionally zero, AA credit. (BA's fares earn 25 percent credit. The full list, by airline and specific fare classes, is on aa.com.) This can sometimes be avoided by booking the same flight under an AA code-share number, in which case AA mileage-earning rules apply, but that may involve a higher fare and can sometimes be done only in conjunction with an AA flight from the U.S. to wherever that flight departs.