We're not network engineers. While we have a basic understanding of the Internet and how it works, we're not blessed with the knowledge required to fast-track your connection to the PlayStation Network. Fortunately, some people are, and there's a nifty trick doing the rounds that will apparently help you to log in – even if you've been struggling for days.
Sony has said that it's "slowly" restoring service, but if that's not good enough for you, navigate to 'Settings' on your PlayStation 4, then click 'Network', followed by 'Set Up Your Internet Connection', and finally 'Custom'. Set all of the values to 'Automatic', apart from the area labelled 'MTU Settings', which you should manually change to 1473. [ Update: You may not need to do this anymore, as PSN service has been restored for many ]
Again, we've no idea why this works, but apparently it does. Remember, you should revert the MTU setting back to 1500 when the PSN is fully fixed – and also, we're not taking any responsibility for any subsequent errors that may occur as a result of this guide. Hopefully, one or two of you will find it of use, though.