Stop users saving to desktop

Denials rarely cause problems. Normally, people either redirect the desktop to a server share or they deny write access.
New program shortcuts normally don't go to the private desktop, but rather to the public desktop (c:\users\public\desktop), so you won't have problems regularly.

Why would want to do this?

It's the equivalent to deny users to place printed papers they work on onto their real desk..

Desktop specifically, or anywhere on the local C: drive? They can also save to my documents or other folders as well.

Really the best choice is to properly train the users. However you could via logon script make the desktop folder in their profile read only.

change the default location of desktop
go to the computer, right-click the desktop shortcut
then change the target location

IMO the best solution is to redirect the desktop (through group policy) to a server location.  You can make it part of a roaming profile if desired, or redirect only the desktop.  I've done this at a number of client locations. When the user complains that it takes a long time to log on, I say,  "Well, you have about 10 GB of files saved to your desktop. We set this up to redirect to a server location to make sure you don't lose these files if your workstation hard drive fails. But these have to load each time you log on, so it takes time."  Most people either stop complaining at that point or move the files somewhere else, either locally, if they're not concerned about losing them, or to a different server folder.