Lately, the term "clade" has been coopted to refer to any structure of a tree that can be conceived without regard to a specific ancestor-descendant relationship.
Where?In a paper on the root of the tree of life, I have seen the term "clan" used (and explained) for what looks like a clade on an unrooted tree and could therefore, upon further research, turn out to be a clade or (if it contains the root) paraphyletic.
The term "clade", the way I know it, does have an ancestor-descendant relationship in its very definition; it's merely not necessary that the ancestor mentioned in "an ancestor and all its descendants" be known.
a paraphyletic clade, which is hardly an inconsistent phrase as this has also seen print with few arguments taken against it.
Because few people bother. You won't see endless laments about the _tuberocity_ in the original description of *Microraptor* or the grammar mistakes in the outstanding paper on the forelimb of *Triceratops* (December issue of JVP).
Would it be consistent, then, to argue that "paraphyletic" and "clade" cannot be used to describe a singular structure?
What do you mean by "singular"? Does "a singular structure" mean "the same taxon"?