When is it considered poor practice to use the static keyword in Java on method signatures? If a method performs a function based upon some arguments, and does not require access to fields that are not static, then wouldn't you always want these types of methods to be static?
Two of the greatest evils you will ever encounter in large-scale Java applications are
These ruin the modularity, extensibility and testability of your code to a degree that I realize I cannot possibly hope to convince you of in this limited time and space.
*A "pure function" is any method which does not modify any state and whose result depends on nothing but the parameters provided to it. So, for example, any function that performs I/O (directly or indirectly) is not a pure function, but Math.sqrt(), of course, is.
More blahblah about pure functions (self-link) and why you want to stick to them.
I strongly encourage you to favor the "dependency injection" style of programming, possibly supported by a framework such as Spring or Guice (disclaimer: I am co-author of the latter). If you do this right, you will essentially never need mutable static state or non-pure static methods.