500 errors ruined my life for the better part of a year, so I can certainly understand the problem. I'm sure you have the same picture in your mind of annoyed visitors/customers each time one of these occurs.
My best success was with my web hosting company's technical support folks. Typically, NOT by chat or telephone, but either e-mail or, preferably, if they have it, some sort of "trouble ticket" system where you can attach files and write info at length in an on-line web form. Hopefully, you won't have to resort to Forums, as they are rarely watched by web host staff in a timely fashion.
Here is what I would look for: a cluster of these 500 errors all about the same time. It is more important that they be very recent, less than 24 hours ago, preferably in the last 8 or even 2 hours.
Cut and paste the errors out of the web logs, if you have access to them, including the time stamps. Attach as a file, if you can. Otherwise, cut and paste into the "report" you write them.
Specifically ask them if they can tell you what is wrong. Typically, they would look in the MySQL database error logs, which customers like you are usually not allowed to see, as it has info on other customers, too. They also have access to a log on the web server's processor usage, which can help, too.
In my case, the web host company had installed proactive monitoring software that killed "everything in sight" whenever the web server processors got busy. That "kill" produced 500 Internal Server Error messages for my users.