Common-not uncommon
significant-not insignificant
responsible- not irresponsible

You could reverse it.

Not common-common
not significant-insignificant
not responsible-irresponsible.

There are so many words in the English language that you can choose from that are like common, not uncommon. Depending on what people are speaking about, they use a specific one. They use not uncommon to emphasise what they are saying, but if it isn't a vital/important part, they just use the simple common. That's what I think, anyway.