Some examples of such properties include 1) lexical fixedness: e.g., neither 'shoot the wind' nor 'hit the breeze' are valid variations of the idiom shoot the breeze and 2) syntactic fixedness: e.g., The guy kicked the bucket is potentially idiomatic whereas The bucket was kicked is not idiomatic anymore; and of course, 3) non-compositionality.
some approaches look at the tendency for words to occur in one particular order, or a fixed pattern.