How do I fix a randomly occurring 500 internal server error (WordPress?)

Unless you have very little amount of RAM available on your hosting server, it may help to run your site through a profiler, such as Xdebug or xhprof and find out your peak memory usage per request and where it's coming from down to the function calls causing it. Chances are you can optimize/disable some of your code to reduce memory footprint per request.

If you're running PHP in FPM mode make sure you're not allowing the master process to spawn more child processes than your server could handle, and never set an insanely high max memory for PHP. Closely monitor your PHP upstream response times. The longer it takes to serve each request, the longer the memory will stay allocated for each process.

There are few methods through which we can fix randomly occurring 500 internal server errors.
1 Sometime it might be due to low memory space left so trying by increasing the space.
2 It might be some issue in .htaccess file
3 Sometime installed plugins are not compatible
4 Check Debug log might be it will give what is the exact problem is there.