This is a very broad question, probably with an even broader answer.

Perl is often called the "glue language of the internet". Its very often used to write small tools that convert data from format A to format B so two different programs can work together. It's used to write small tools to extract data from some files, or do some small database job. Or to add a logo to hundreds of pictures. Or to automatically spam your friends with birthday wishes so you wont have to remember.

Perl is also used for big tools and major projects. Implementing an internet search engine, running major website, working as part of a multi-million bookkeeping package, doing database maintenance for major banks, mapping the human genome or running business critical tasks for NASA. You name it, someone has probably done it.

Some of those you'll find mentioned on the internet, some you will never hear of unless you work in IT for that specific company or organization. You can certainly look through the PerlMonks categories Cool Uses For Perl and Meditations to find people mentioning how they use Perl and the cools things they did with it.

But i think your question is fundamentally flawed. You don't take a solution and try to find a problem for it (e.g. trying to find applications for a specific software technology). That is why so many business falter.

What you do is find an interesting problem and then find the right tool(s) (and people who can work with them) to build a solution.

You know, "when you only got a hammer, every problem looks like a nail", that sort of thing.