On ual.com the only award tickets you can see are those on United metal. Even searching for award tickets on United metal can be a painstaking process. The best way to find award tickets is to call United and also have a knowledge of their *A partners.
As for the best awards I'd say go for SQ F, NH F, LH F, SA F, or TG F.
Obviously it would be wise to look for the premium routes on the above carriers to best take advantage of what they have to offer.
SQ: Something via SIN and on the Megatop (744). I know there is the new 77W product but getting that on award at the moment is near non-existent. Another must is a visit to the SKL in SIN.
NH: New style config routes. Currently on Narita-Europe/UK and North America Routes.
LH: A route that will allow you to experience the LH F Terminal in FRA and the services associated with the LH F Terminal.