If you're as uptight and as obsessive compulsive as this particular author, you'll no doubt hate the sight of zero per cent Trophy sets on your PlayStation Network profile. Fortunately, you can now remove them on your PlayStation 4 courtesy of firmware update v2.50. Isn't the present a magical place?
From the PS4's home screen, push up on the DualShock 4 to reach the top layer of the system's menu, and scroll right to the cup labelled 'Trophies'. Select it with X on the controller.
You'll be presented with a list of games across all PlayStation platforms. It's worth noting that you can only remove Trophy sets that are stuck at zero per cent, so you'll need to find one of these before you can delete it. Scroll through the list to the title that you want to remove, and hover over it.
While hovering over the Trophy set that you want to remove, push the options button on the DualShock 4 and a side menu will appear on the right-hand side of the screen. Scroll down to 'Delete' and select it. You'll need to confirm the process first. Click 'OK', and breathe a sigh of relief.