Though I prefer Saké on its own, it is also a versatile drink that can be used in many different cocktails. As Martinis are currently very fashionable, you are starting to see Saké used in a number of different Martinis. At its easiest, Saké can be used as an interesting substitute in almost any cocktail that calls for vodka and gin.
For example, you could make a simple Saké Martini, substituting Saké for your vodka or gin. Or try a Saké Margarita, Saké Screwdriver, Saké Sunrise, Saké Sour or a Saké Bloody Mary. Such cocktails do not require any other changes except substituting Saké for either main liquor.
In general, if making such a cocktail with Saké, I would recommend that you use Junmai or Honjozo Saké. I would not recommend using a Ginjo or Daiginjo Saké as they are generally more subtle and in a cocktail you would lose many of those flavors. A Junmai or Honjozo are more likely to stand up to the other ingredients in a cocktail.
Nigori Saké is a special case as it tends to have a milky color and be sweeter. It would not be a good substitute for basic cocktails as mentioned above. But, a Nigori would go well in a cocktail that is intended to be sweeter and in which the milky color does not detract. A Nigori Pina Colada might work well or maybe mixing it with some Bailey's to make more of a dessert Martini.
Beside the basic cocktails, restaurants and bars are getting creative with Saké, concocting their own special creations. For example, Haru in Boston has their own version of the Saketini, which contains vodka, Saké, and plum wine. It also has a few thin strips of cucumber in it. This is quite delicious and was not overly sweet.
You can find plenty of other Saké cocktail recipes online. Just do a Google search for "Saké cocktail recipes." But I do want to mention two other intriguing Saké concoctions.
What appears to be an American creation is the infamous Saké bomb. This is not really a cocktail as such, but more just a specific way to drink Saké and beer. You start by filling a beer glass halfway with a Japanese beer like Kirin, Sapporo or Asahi. You could use other beers as well but it is more thematic to use a Japanese beer. Then fill a small Saké cup with warm Saké. Place two chopsticks in a "V" shape atop the beer glass. Carefully balance the Saké cup atop the chopsticks. You then slam your hand down on bar so that the Saké cup falls into beer and you immediately chug the beer and Saké mix. Just be careful not to get hit in the face by the Saké cup in your beer glass. To me, it is a waste of good Saké but it is popular with some.
Another interesting recipe is Saké Sangria. This makes more of a white Sangria. You mix 1 1/2 cup Saké, 1/2 cup orange juice, 1/4 cup brandy and 4 teaspoons of sugar. Mix thoroughly until the sugar is dissolved. Then add some mixed fruit and top it off with a little soda water. Place it in the refrigerator and let it sit for 4-6 hours. It is then ready to drink. I have not tried this yet but it sounds interesting and I will probably try it soon.
If you enjoy Saké, then be adventurous and try creating your own cocktails with it. If you create something really good, feel free to post it here in the comments.