As a hint in finding eligible fares (H and above). If you go to united.com and book a flight (don't use the one on the home page, click on "More Search Options". On the next screen, in the 'More Search Options' area, there is an option on 'Upgrade Eligibility.' Change that to 'Miles-MH', and it [i]should in theory[/] bring up H fares. These are UA's lowest fare buckets for upgrading with miles. It doesn't tell you that an upgrade is available, but it does search for eligible fares. YMMV