Is the song "Hooray For Hollywood" in the public domain?

I very much doubt it. The song was written in 1937, and the composer, Richard Whiting died about a year later. However the lyricist, Johnny Mercer (others were involved as lyrics changed through the years), lived until 1976, However, the STUDIO for whom the song was written for one of it's films, is still around, and they probably owned the rights in the first place. As it is, you can't just go by age of song or even death of the original creators of the song---copyrights can and are frequently renewed. The song (usually played as an instrumental version) is still used as part of the Academy Awards ceremony (and they might have bought the rights to the song outright).

I can't say for sure who has the current copyright on this (you can find out if you absolutely need to) , but I'm pretty sure it's not in the public domain. ESPECIALLY recorded versions of the song.