If you're having trouble staying signed in to Pandora, make sure you don't have a setting selected that deletes your cookies or browsing history on exit. (Your sign in used to be stored in Flash local storage and on the new site it's stored in cookies.)
If you're using the following add-on/extensions: Better Privacy or Flashblock, make sure to allow Pandora within them, so you won't be signed out automatically.
If you're using C Cleaner or ATF to remove temporary files from your machine, this could also be affecting your ability to stay signed in.
Lastly, make sure your security software, such as Norton Internet Security, isn't automatically performing "cookie cleaning" which might cause this.