How not to solve P=NP?

There are lots of attempts at proving either $\mathsf{P} = \mathsf{NP} $ or $\mathsf{P} \neq \mathsf{NP}$, and naturally many people think about the question, having ideas for proving either direction. I know that there are approaches that have been proven to not work, and there are probably more that have a history of failing. There also seem to be so-called barriers that many proof attemps fail to overcome. We want to avoid investigating into dead-ends, so what are they?