LakeCentralFlyer

What is the best way to book multi-day stopovers/search for fares that allow this?

MSPeconomist

When you do multi city search, you should be given the lowest applicable fare, whether is is pricing by segments or as a RT with a stopover. Does it matter which way the price has been calculated as long as you get a low price.

Especially if you have status, you could call the airline and ask which of their fares allow stopovers and what the rules for stopovers would be. Often the free stopover must be on the way, at a hub or natural connection point, so check legal routings. Other fares are based on MPM, so you might be able to get any stopover you want as long as you don't fly too many miles.

dogcanyon

I have used ExpertFlyer to find fares that allow stopovers, then used the Orbitz multi-city option to book them. Last summer I booked a Singapore to Berlin fare on Royal Jordanian this way. The fare allowed stopovers in both Hong Kong and Amman and allowed Cathay Pacific as the carrier between Singapore and Hong Kong (since Royal Jordanian does not fly that route).
It can be difficult to make all the planets align and get the routing/carriers/booking classes that you need on a multi-stop, multi-carrier itinerary for the fare (in my case the total one way fare was about $690).
Often Orbitz would initially display a routing with the fare at around $4,500,
then I would select the fare and when the next screen appeared it would say "Congratulations, we've found an even lower fare!" and then show it at
$690. Good luck - it may take some coaxing of the booking engine to get what you want.