What is more common, Duch Święty or Święty Duch? (In Gdańsk there is ulica Świętego Ducha)
Definitely "Duch Święty". I'm quite surprised that there is such a street, looks like a rare exception, may possibly be quite old. In Poznań there is "ulica Święty Marcin", which is even more surprising as technically it violates grammar rules - normally you have the street of someone, so that would be "ulica Świętego Marcina".
Holy Spirit is rejected an answer without a "The", but a context is needed to decide on an article. A prayer could begin "O Holy Spirit...", where no article is appropriate.