The original question is confusing, and in any case the answer is usually "it depends" as in it depends on the type of ticket: Y/F, domestic/international, peak winter/summer, etc. We'll assume you are asking about the lowest "saver" award levels, not the mid-tier or anytime awards.

In general AA has the best availability across-the-board domestic but still can't get me to Mexico at Xmas. AS never seems to have saver awards to Mexico when I need them or on any dates remotely close. UA has been surprisingly good at domestic F seats but tougher for Y and impossible for TATL or TPAC in C (on UA metal). UA and AA probably come closest to finding seats close to my preferred date on the fare calendar. UA and WN (RR 1.0) do very well on last minute bookings, which are useful for bowl games and funerals but not for school vacations.

I don't think I really have an airline that comes "close but no cigar" Either there is availability within a couple of days of when I need it or there is nothing the entire month.