The simple facts are these.

Paid Travel
1
DL gives free upgrades to Biz, depending on status and you can boost status with the Amex Reserve card and get into the lounge (drinks free).
2
AA makes upgrades to Biz easy with miles and a small fee. If you time it right, you can get to Gold (silver equivalent) with one transcon trip by a Gold Challenge.
3
UA gives you economy plus ALWAYS, but upgrades are more expensive, and less likely free. You can buy economy plus for the first year or get it with a Chase card and then not need it after that. You can also get upgraded rarely on CO and accumulate on US flights as well.


Redemptions AKA Free travel
1
AA blocks BA TATL, but gives one way awards and has a great award calendar.
2
UA
UA is worse, but has great *A partner awards but then blocks *A awards randomly.
CO
You can accrue on CO, but fly UA with buying e+ for 1 year, then the reciprocal upgrades for CO on UA should at least get you free e+ from 2011. You can fill CO with Chase visa and Amex MR, and the awards are easier to get than UA.
US
US is like DL, for upgrades (free, but almost never) and has open *A awards but high fees.
3
DL has bearable domestic, but terrible Intl awards and partner awards.

All 3 are now searchable on expertflyer, 5$ a month.