The first, is the phrase "Simon sez" or "Simon says"... if this were an idiom, it would cause major problems for many theories of idiomaticity because it appears to be an agent-verb idiom (contra Marantz and others). I question this phrase's status as an idiom for a number of reasons. First, it's use seemingly requires a call-back to the childhood game. Unlike other unambiguous idioms, it is impossible to describe its "meaning"without referring to the game; whereas an idiom like "kick the bucket" is easy to describe without referencing the real world. Second, it lacks a real effect on the semantics "Simon sez get out of the car" and "Get out of the car" are roughly identical statement. Whatever its effect is, it is pragmatic. Now, neither of these arguments are knock-outs, but it does seem to me that "Simon sez" belongs in a different class.
The second controversial phrase was "Does the pope shit in the woods?". This phrase is apparently used as an obvious affirmative to a yes/no question. It was featured in the Big Lebowski. It is a play on the phrases "Is the Pope Catholic?" and "Does a bear shit in the woods?". I am less certain about this one, but I am still disinclined to describe it as an idiom. For one, it is obviously used to be funny. It is well known that we can play with expression and grammaticality in language for humorous effect. This phrase seems no different. To me, idioms require a certainly level of casualness in use that is not found with this phrase. To use an idiom, a speaker must almost be unaware that one is using an idiom barring reflection. [There's got to be a better way to put this.] Further, I believe that this phrase is a play on two common, but non-idiomatic phrases. There is a real sense in which the phrases "Is the Pope Catholic?" and "Does a bear shit in the woods?" are used compositionally when used as answers to yes/no questions. Their answers are trivially yes, and are meant to express just that.
Obviously, there is a lot of grey areas in idiom studies. Expressions like these do illustrate the need for a formal definition of idiomaticity. Thoughts?