While LAX has A380s it's a far cry from the 12(?) different carriers that flew them there previously.

There are plenty of search tools, however most of them require payment to get the best searches (primarily searching multiple days and far in advance). If you want to focus on one carrier, I'd recommend Etihad. They only fly to LHR and JFK (I believe), so book to JFK so you get more than 6 hours on one. Amex transfers to Etihad Guest, while more expensive after the recent devaluation, their own program has the best availability.