Depends on age and cultural background. When I deal with non-technical folks of various ages, there's an OK (but not great) chance that they've read some classic literature that I enjoy. Unfortunately a lot of those were forced readings in high school, but it definitely depends on the individual.

As much as I love Star Trek (particularly TNG), I'd say it's far, far less popular among folks in their early 20s these days than the classics. Probably why they're called classics :)