Technically not regular expressions since they are appropriate for matching much more than regular grammars, but that's true for almost any programming language's "regular expressions".
IMO the word "improvements" should be in quotes. Pathological regular expressions are an issue because of things like back references and look-behind matching. If Perl hadn't enshrined the idea of "a regular expression for every problem" people would probably be less skeptical of regular expressions in code.
(The extensions to regexps in Perl 5 (and presumably now also in PCRE?) have made it possible to write them more like grammars than anything else. But you would probably not approve. :-) )
Make syntax consistent across RE engines for one.
Improve the syntax.
Introduce clear modes with different tradeoffs, so you don't get e.g. regex based DOS attacks.
Make them easier to compose, not just through copy pasting...
The real sweet spot I've found is parse in Rebol / Red - http://blog.hostilefork.com/why-rebol-red-parse-cool/ | http://www.rebol.com/docs/core23/rebolcore-15.html
In general you have the same problem when parsing CSV, command line flags, URL query parameters, JSON, etc.
(For those having the type-safety problem with JSON in Python, can I recommend my own library obiwan? https://pypi.python.org/pypi/obiwan/ :) )
[0] https://hackage.haskell.org/package/attoparsec
[1] http://www.serpentine.com/blog/2010/03/03/whats-in-a-parser-...
It's pretty fast too. I don't know whether you can compare the performance of Ragel to regex libraries like RE2; they look like they solve different problems.
"Parser combinators", for me, denotes actual parsers, for non-contextual or contextual grammars.
For the little history, I started this for URLs[1]. I couldn't (and still can't) figure out a proper interface for that, but someone mentioned he would like something like that for regular expressions, so here it is.
We already have something similar for command line flags in OCaml[2] (and several for JSON).
[1]: https://github.com/Drup/furl [2]: http://erratique.ch/software/cmdliner