I asked someone in Moscow what a suitable translation is for "urban legend", and he said that байка would work (this person is young and didn't understand a literal translation like городская легенда). But he thinks байка is more often used for events that happened quite recently, say in the last year, and that isn't a constraint on the term urban legend. The Wikipedia page for байка doesn't suggest that there is a very narrow temporal constraint on its usage. Can байка be applied as broadly in time as urban legend?
Байка - is not quite the same as urban legend. Байка is much closer in meaning to word anecdote (in its English meaning, not Russian) and originates from a Ukrainian word meaning басня in Russian (apologue in English).
While words байка, басня and выдумка (made up story) can be used, they are significantly more colloquial in their meaning. I don't think there is a word (or a stable phrase) analogous to urban legend, but if I had to translate it, I would probably use word миф (myth).
Байка is a story, but "Urban legend" can be also a well-known "fact". For instance one of such "facts" is that cactuses protect from computer monitor's radiation. Or if the bullet hits the tank with fuel it will immediatly blow up.
The translation for "Urban legend" is городская легенда. I found only two examples in Russian National Corpus:
Несколько девушек, видимо, слышавших городские легенды о распространителях вируса, колющих посетителей в ночных клубах, пронзительно завизжали. [Герман Садулаев. Таблетка (2008)]
Так, осененная всевозможными литературными байками, городскими легендами и родительскими фантазиями, дочь вошла в мир. [Владимир Арро. Дом прибежища // «Звезда», 2002]
Also, Wikipedia article entitled Городская легенда has a link to an scientific article that uses the "Городская легенда" in its title:
Е. В. Смирнова. Устойчивые мотивы сюжета «геопатогенная зона» в городской легенде (2010).
This is a newspaper article that uses this term:
So, I think that the term is relatively new, but it is used nowadays.
Urban legend is a term of USA origin and obscure semantics. Городская легенда is obviously a verbatim translation of it. It is very recent and not well-known. Before the term was introduced in Russian language, Russian urban legends had been called миф or легенда or whatever -- without the need for 'urban' connotation. So you can readily go with these two (I'd prefer миф for its supposedly fake nature). If you use городская легенда, you have less chances to be understood correctly or you may sound "westernized".