I've covered this before, but in light of a recent experience, I believe it's worth mentioning again: every major US airline is a member of one of the three airline alliances (Star Alliance, Skyteam, and Oneworld). So if you can't find the tickets you need, don't forget that your miles aren't just good on one airline! This is particularly important to remember for international travel when your carrier of choice may have limited overseas routes or doesn't fly to the city you want to visit.
Alliances also often ofter a better value for your miles. For instance, I'm currently booking tickets for my parents on Delta. Delta's "Saver" TATL Business class flights are currently going for 125,000 Skymiles. If there are no "Saver" awards available, you could be looking at paying 200,000-350,000 miles for those tickets. But Delta partners with two very versatile TATL carriers, Air France and KLM. And with Delta, partner awards (if available) always book at the "Saver" level.
The Oneworld Alliance recently came through for me in a big way, thanks to the recent US Airways/American Airlines merger. A few years back I acquired the British Airways Visa (with a 100K mile signup bonus!) with the idea of flying in BA First with Mrs. Pointsninja. That was before I realized how high the fuel surcharges on BA (a Oneworld member) were. Yikes! I ended up using those miles for alliance awards( with Oneworld member LAN) on our Peru trip.
Well, US is now a member of the Oneworld alliance but, as many bloggers have noted, US doesn't seem to have its award software fully integrated with the Oneworld yet. As a result, US isn't currently imposing fuel surcharges on BA flights. What that means is that by using my US miles I was able to book MGM-DFW-LHR-FRA (roundtrip) in BA First for 125,00 miles and $192 in taxes and fees. That same ticket booked with AA or BA miles would have cost me the same number of miles, but over $2,000 in taxes and fees!