There's still a lot of debate around this topic. It's obviously difficult for people who have used these methods for the past 60 years to simply say "I've been using a flawed method for 60 years" – although in the end that's how science works. The problem moreover is double: the method has built-in flaws, and on top of that it's often misused.

Some starters:

• The official statement by the American Statistical Association

• A follow-up editorial

• Signatories for the dismissal of the method

• Many papers explaining the built-in flaws, from this old 1935 paper and this old 1965 discussion, to more recent ones; for example this, or this, or this, or this, or this tutorial

• This paper gives a good summary

• Journals that don't accept "statistical significance" methods anymore: this or this

• Several books, for example this one. I agree with the factual content of this book, but I don't like the authors's braggart way of writing. In their defence, though: it's the same braggart way of writing that R. A. Fisher, the father of "statistical significance", often had.

What's sad is that these discussions easily end in political or "football-team"-like debates. But the mathematical and logical proofs are there, for those who care to go and read them.