I get questions all the time like this:
Hi, I have a million American Express Membership Rewards/Chase Ultimate Rewards/Citi ThankYou Points, what is the best transfer partner to go from [here] to [there.]
There are basically two ways to answer the question:

Look at award charts of all the points' transfer partners and then look for award space using the cheapest miles. I used this method to suggest to my friend that he transfer his Membership Rewards to Flying Blue miles to get from Denver to Tel Aviv.

All Nippon Airways (ANA) has reduced fuel surcharges drastically on awards that travel to or from Japan. Now you can fly to Tokyo from the United States for 75,000 miles in flat bed Business Class roundtrip plus low taxes plus a $4--yes four dollar--fuel surcharge. That's tens of thousands of miles fewer than competitors charge.

The above screenshot is of a roundtrip Business Class award flying ANA between Los Angeles and Tokyo in February 2017.
Routing Rules
ANA has two award charts.

American Express just added Etihad as a Membership Rewards transfer partner in April, and it is now offering a 30% transfer bonus through June 15, 2016.

 

For the next six weeks, 1,000 Membership Rewards will transfer to 1,300 Etihad Guest Miles.
Good Deal?
Using Etihad Miles for Etihad Flights
Etihad Guest Miles aren’t very good for booking First Class on the Etihad A380, which is a current obsession of mine.

For instance, Etihad would charge 88,022 Etihad miles + $175 to book Abu Dhabi to London in First Class on the Etihad A380.

Through March 10, 2016, American Express is offering a 50% transfer bonus from Membership Rewards to Virgin America Elevate Points.

The normal transfer ratio is 200 Membership Rewards to 100 Elevate Points, but for the next month, the ratio will be 200 Membership Rewards to 150 Elevate Points.

How Membership Rewards Work
As a quick refresher, there are two ways to use Membership Rewards:

The dumb way: use them like cash to purchase flights, car rentals, hotels, or cruises at a paltry rate of 1 cent per point
The smart way: transfer them to one of the 16 airline partners

I use united.com for most of my award searches when redeeming Star Alliance miles because it shows most Star Alliance partners' award availability, but it does have some drawbacks. Namely, it doesn't show all partners and it sometimes displays phantom award space--award space will show as available but in reality, it doesn't exist.

Star Alliance Airlines Whose Award Space I Want to Search

The best way to confirm Star Alliance award space is using the All Nippon Airways (ANA) search tool. ANA's tool is trickier to use, but it's the most reliable.

Effective this Thursday, October 1, 2015, 250 Membership Rewards will transfer to 200 British Airways or Iberia Avios. That's a 20% reduction of the current 1,000 Membership Rewards to 1,000 Avios transfers.

Source: Travel with Grant

This is a huge blow to Membership Rewards. It still has great 1:1 transfer partners like Delta, ANA, Aeroplan, and Singapore, but this majorly hurts the value of its best oneworld transfer partner.

In the past we've seen transfer bonuses from Membership Rewards to Avios like this year's 40% bonus.

I'm going to show you how to book a mega-award with seven stops to South America with Asia Miles. You already have Asia Miles because they are a transfer partner of Citi ThankYou Points, AMEX Membership Rewards, and SPG Starpoints.

Asia Miles can be used to book any oneworld airline's award space plus space on its partners Aer Lingus and Alaska Airlines.

Asia Miles charges fuel surcharges on award tickets when the equivalent cash tickets would have fuel surcharges.

American Express Membership Rewards transfers to Virgin Atlantic miles will come with an automatic 25% transfer bonus from now through April 30, 2015.

Well, it's January 31. So today is your last chance to transfer. Since I posted this initially, I have two other posts on Avios:

New, Worse Avios Award Chart takes effect in April.