How do I handle nasty comments from customers on the software that I built

As @hatkyinc points out, it's them not you. That's the way some people are when they encounter a problem.

In your role as customer support you need to show empathy without becoming emotionally involved in the situation yourself. That's the best way you can help them.

It's about them not about you. Someone killed thier pet earlier in the day or something, who knows, bad childhood, taken advantage off, never learned more than the first 1,000 words in school, ignore the language.

Only counter to software issues I know is tracking and testing tools. Backend and frontend APM reporting, automated testing that includes blocking normal things like JS or some of your subdomains or 3rd party subdomains, especially if it's for businesses. There are tracking tools that lets you reply full sessions, all mouse movement and keyboard and sometimes even show rendering gaps (most won't, but you might notice clicking at the wrong place reloads the page). Anything can and will break at some point, silly filters, incompatible browsers, software and devices that act differently just on that one little thing. Out of memory and network issues. Even if their home electricity is spiky it would be your fault your software renders weird lines on the screen.