I watched It Came from Outer Space today for the first time since childhood. One of the best shot 3D movies of the early 50's it is also one of the better scripts written largely by Ray Bradbury.
The score used work in equal parts from three composers: Herman Stein, Irving Gertz and Henry Mancini which were combined for the soundtrack and featured Bud Hoffman on theremin and is credited as one of the first films to feature it. I haven't found yet which of them wrote the theremin cues.
There is a lot of melodic theremin parts featured in the opening titles and in the alien POV and attack scenes.
The iconic creature POV image is featured as the TW error page (which hopefully you've never seen).