Start the game, black screen, the "Hello Games" logo comes up, then the game crashes to desktop with a "No Man's Sky has stopped responding" error message.

Same. All we can do is wait for a patch. So long as you meet minimum specs waiting is all we can do. :(

Got AMD Phenom or older Intel Processor? Same here :-/
Currently the game is compiled against SSE4.1 and crashes if the CPU doesn't support it.

You can check if that's the problem if you run the Intel SDE:
https://software.intel.com/en-us/articles/intel-software-development-emulator

Download the Windows version, unpack and run that batch file. From the opening command prompt cd you way to the NMS binaries directory and run NMS.exe.

In my case the game works that way (very slowly...).
Nothing we can do other than wait for a patch...