Currently Turkish Airlines (a Star Alliance member) has widespread availability in business class to Istanbul and beyond. For many dates you can find as many as 8 business class seats available per flight which is rare.
How to Search
The best and easiest way to search for this space is on the United Airlines website by using the flexible dates tool. It will show you a calendar for a two month span and indicate which days have premium cabin availability. You can also use the Aeroplan website although the interface isn’t nearly as intuitive.
The two easiest programs to book Star Alliance awards for most U.S. based travelers are United MileagePlus and Air Canada Aeroplan. Aeroplan is a transfer partner of American Express Membership Rewards and charges 57,500 each way in business class from North America to Europe. MileagePlus is an Ultimate Rewards partner and charges 70,000 miles each way for this same redemption.
For more info about routes and availability, see: this View from the Wing post.