It is clear that there are lots of problems that look like a simple regex expression will solve, but which prove to be very hard to solve with regex.
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At the very least, I'd say learn regular expressions just so that you understand them fully and be able to apply them in situations where they would work. Off the top of my head I'd use regular expressions for:
Regular expressions at a theoretical level form the foundations of what a state machine is -- in computer science, you have Deterministic Finite Automata (DFA) and Non-deterministic Finite Automata (NFA). You can use regular expressions to enforce some kind of validation on inputs -- regular expression engines simply interpret or convert regular expression patterns/strings into actual runtime operations.
Once you know whether the string (or data) you want to determine to be valid could be tested by a DFA, you have a choice of whether to implement that DFA yourself using your own code or using a regular expression engine. You'll find that knowing about regular expressions will actually enhance your toolbox and your understanding of how string processing can actually get complex.
Based on simple regular expressions you can then look into learning about parsers and how parsers work. At the lowest level you're looking at lexical analysis (where regular expressions work) and at a higher level a grammar and semantic actions. These are the bases upon which compilers and interpreters work, as well as protocol parser implementations, and document rendering/transformation applications rely on.