What is an example of a modern English clause that does not follow the verb-second (V2) word order?
V2 certainly is a technical term in linguistics.
We use it to refer to the word order of languages such as German, which require the matrix verb in a sentence to be the second constituent of the main clause. V2 word order is the reason that in German, subordinate clauses seem to be verb-final, while main clauses seem to be SVO.
English has stylistic inversion, which resembles this behaviour (e.g. little did he know), but this may be another phenomenon altogether.
I am a bit puzzled by your question, though, because modern English no longer has this in unmarked contexts: compare Yesterday I read the book with Dutch Gisteren las ik dit boek.
V2 (which yes is a linguistic term but is fantastically common in the literature on Germanic syntax) does still have minor minor traces in English, but your question phrases the idea that we would need to search hard to find a non-V2 structure in PDE (Present Day English), when precisely the opposite is true. Many people would even say it has completely disappeared, but I have exposure to quite a lot of language from around the world because I work for a transcription company and these things stick out to me. I actually noticed one today, I'll grab it now:
Then, of course, on top of a tiny Norman tower was put this enormous spire. (Said by a man in the south of England, approx 50ish, giving a tour to a visitor)
Here [on top of a tiny Norman tower] is functioning as topicalised constituent which is where you see typical Germanic V2 occurring. It's also quite common in certain negatives when sentence-initial (and arguably the last occurrence where it is recognised by most speakers).
Never in a million years [would] [I] think of doing that.
Not until she apologies [will] [I] talk to him again.
Basically, when talking about V2 in English and other Germanic languages, questions aren't really where you look and especially in a generativist reading these are not even the same thing. Yes-No questions involve auxiliaries which undergo T-C movement and it's not the case of any functional force needed that stipulates that the verb has to be in the second constituent position.
So, non-V2 patterns are the absolute standard in PDE.