Is there a way to save an MP3 from a web server directly onto an iPad? I can stream it through the browser, but there is no obvious "save" capability.
Would it make a difference if the MP3 were an email attachment?
As fas as I know there is no way to save an MP3 file directly from the web. What you can do is jailbreak your iPad and then try saving it.
You could save it into an app, such as AirSharing, then use iTunes to save the file to your computer, then put it into the Music Library like normal.
There is unfortunately no direct way to get a file contained somewhere on the Web, or even in an app, into the music library/video library, without first using iTunes.
Same goes for having it e-mailed to you. You can always access the e-mail and re-stream the MP3, but note the emphasis on that. Continually Stream the MP3.
You must use iTunes to get it into your library without any other significant process, such as Jailbreak as noted by Chirag.
GoodReader is what you're looking for. Copy the URL into it or use the built in browser to find the file you want then download it.
My favorite is Documents by Readdle. It has a builtin browser.
You can simply browse your desired webpage and download different kinds of files like mp3, pdf and others into your iPad directly and enjoy playing or reading them in the same application.
It is possible, by jailbreaking the iPad and then installing a tweak called Safari Download Manager. This will allow you to save any file from Safari to your iPad.
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