It has some minor uses, but not a whole lot. If, for example, you wanted to do recursion with an anonymous function, you could use the Y combinator (well, more likely the Z combinator which is a variant of the Y combinator that works with strict evaluation). However, it'd probably just be easier (and clearer!) to give the function a name and do it that way.
However, as someone pointed out, the Y combinator's utility is not as an actual tool for programmers but as a tool for mathematicians working in computational mathematics. Alonzo Church solved the Entscheidungsproblem using a primitive variant of the Y combinator called the omega combinator. The Y combinator itself was invented by Haskell Curry (his name is familiar for a reason) in order to demonstrate what is commonly called Curry's paradox.
However, as someone pointed out, the Y combinator's utility is not as an actual tool for programmers but as a tool for mathematicians working in computational mathematics. Alonzo Church solved the Entscheidungsproblem using a primitive variant of the Y combinator called the omega combinator. The Y combinator itself was invented by Haskell Curry (his name is familiar for a reason) in order to demonstrate what is commonly called Curry's paradox.