For general-purpose applications (e.g. use in programming languages) people usually want a regex flavor with support for at least backreferences and typically also (recursive) subpattern references. Note: Adding support for backreferences makes matching regexes an NP-complete problem (as opposed to a simple linear-time algorithm without them).
(But of course, having fast NFA implementations for really-regular regexes is still useful. After all a large part of the regexes you typically write will not use backreference etc)