I think that it sounds pretty much the same but I could see those nunances:
Common: It's what people would usually do. Like common knowledge, knowledge that everyone has. When you say something is common, it's something most people would probably do.
Normal: it's the standard way, how things were supposed to be. If we take a video game with different levels of difficulty, "normal" would be the normal choice, but people will commonly pick easy or hard. So if you say something was normal, it was "how it was supposed to be", I guess, with no surprises.
Ordinary: It's something that happens pretty much everyday. Again, it can be related to you or in general. Like, ordinary people have breakfast in the morning, nothing surprising there.
So I guess you could switch them pretty much anytime you want and it would make sense. They all have that "unsurprising" feeling.
To my hear, the sentence: "I think this phrase is very... common" is correct because it conveys the meaning that most people would use that sentence.
Hope it helps, as always, this is what I think. maybe I am wrong ^^