Is the Itsy Bitsy Spider public domain?
The traditional song is generally treated as public domain: it has no known author and goes back to at least 1910. A particular recording, video or arrangement can still be protected.
What’s usually safe
Singing it at home, in a classroom or at a story time is ordinary use of a traditional song. If you’re recording it, the safest route is your own voice and your own simple arrangement.
What to watch out for
Someone else’s recording, video, illustration or written arrangement can have its own copyright even when the old song doesn’t. Music libraries say this plainly: one notes that the melody is out of copyright, while its own arrangement and production must be licensed. Added verses may also be much newer than the song.
This isn’t legal advice
Copyright rules differ by country and by date. The U.S. Copyright Office says the term for works published before 1978 depends on several factors. If you plan to sell something that uses the song, ask a copyright lawyer first.