popularity-contest is a hard tag to create questions for. Its hard balance between vagueness and broadness. What are some good examples that other challenges should seek to emulate?
The best fit I've seen for popularity-contest recently is this challenge. It's a language-design challenge, but the aim is to find the most appropriate way to work an interesting property into the language. That's something that's a) very difficult to define objectively, b) will have a reasonable consensus upon voters when defined subjectively, c) is nonetheless clearly in the general topic area of programming, with programming and language design skill being very relevant, and d) is likely to be close to the vote counts that the posts would "naturally" get if the challenge didn't have a win condition.
This allows us to extrapolate as to what would make a good popularity contest in general.
The first thing to note is that the victory condition you'd "actually want" has to be close to the typical uses of upvotes, or else you'll get mixed results as some people vote the way you want and some people don't. In particular, you must therefore be looking for elegance, humour, appropriate language choice, golfiness (to a small extent – this is better measured objectively), unreadability/obfuscation, or good explanations. (Note that the reason voting as a sorting criterion works fairly well on many other Stack Exchange sites is that "good explanations" are the "victory condition" there. Even here, tips has a sort of inherent "popularity contest" victory condition, except when it's asking for tips for improving a specific piece of code, which can be more objective.)
Another thing you want to make sure of is that the challenge is still ontopic for the site. If there were a challenge "write code that draws a pretty image", popularity-contest would clearly be the best possible victory criterion. That doesn't make the challenge a good popularity contest, though, because it's a bad challenge, and giving it an appropriate victory condition won't change that. It's important to pick a topic for which writing a popular answer will take skill (especially programming skill), and where the subject matter isn't too far from the sort of subject matter normally covered on this site.
Finally, popularity-contest is a poor victory condition if you're looking for something that can be measured objectively. That's why we don't normally use it for golfing challenges. (We could, I guess – it'd certainly make competition between languages more fair – but it runs into the problem that even in the same language, a better solution submitted later may well end up with fewer votes than a worse solution submitted earlier, so comparing within each language objectively works better.)