Go to ual.com and click 'update profile' under planning travel.

Within display preferences, change availability format to 'an expert.'

Log into ual.com if you're not logged in already, and click 'more search options' on the home page. Click 'multi city.' Enter the appropriate cities and dates, and select 'search by schedule' and 'sort by departure time.'

Flights should now have fare class availability displayed. For instance, F0 Y0 ... would indicate that United is selling no more first class or economy class seats; the flight's being oversold would be a good bet, too. Keep in mind that the maximum number displayed alongside fare class is 9, i.e. F9 Y9, even if United is offering more than 9 F class or Y class seats for sale.