Are you the proud owner of a shiny new PS4? Great, isn't it? At least it is if you haven't fallen foul of the "blinking blue light" problem that some new owners have been reporting.
The blinking blue light issue appears to relate to the console failing to send an audio-visual signal to the television. The console should start up with a white light to show it has powered on; the blinking blue light indicates a problem of some description.
The apparent failure of some launch PS4s has been related by some to the Xbox 360's now-notorious "red ring of death" problem that plagued a number of old-school systems, and has attracted the now-inevitable Amazon review-bombing campaign. At the time of writing there are 611 one-star reviews compared to 1,414 five-star reviews -- not enough to drop the PS4's overall rating below a four-star average, but enough to be noticeable.
Sony's issued a full troubleshooting guide to help those affected to figure out what the problem is. It usually boils down to one of four issues, apparently: TV compatibility issues, problems with the PS4's power supply, problems with its hard drive or an issue with another piece of internal hardware.
To test whether or not your TV is causing the problem, Sony suggests you update your TV's firmware, assuming it's a model that actually supports upgradeable firmware.
To troubleshoot the power supply, turn off the PS4 completely by holding the power button for at least 7 seconds or until it beeps twice, then disconnect the power cord from the electrical outlet and check the cable, connector and power input on the back of the PS4 for obvious damage. If you have a PS3 or other device with the same type of AC cord, Sony also suggests trying that cable to see if it resolves the issue.
To troubleshoot the hard drive, turn the PS4 off completely as described above and disconnect it from the power. Slide the hard drive bay cover -- the left panel on the top of the PS4 -- to the left and remove it. Then check the hard drive is correctly seated in the hard drive bay; it might be worth removing and reseating it completely to make sure it is correctly attached -- alternatively, you are free to replace the built-in drive with your own if you so desire. A replacement drive must be at least 160GB in capacity and no more than 9.5mm thick; full details here.
Finally, if you're still having issues, try booting your PS4 into Safe Mode by turning it off, then pressing and holding the power button until you hear two beeps: one when you initially press, and another 7 seconds later. This will boot into 480p graphics mode and allow you to update the system software, restore the settings to factory defaults, rebuild the system's database of content, format the system and remove all content except firmware updates, or format the system and remove all firmware updates.
If all these steps still don't fix your problems, you'll need to contact Sony. Sony specifically recommends that you try the live online chat option rather than phone, as wait times will be significantly shorter.