Apple - Are there Greasemonkey scripts for Safari?

Another option to run GreaseMonkey scripts is NinjaKit (which is what I use). It's a Safari 5 extension, which means it's just a tad safer than GreaseKit.

I use TamperMonkey and it works surprisingly well. It's also a Safari Extension.

I have tried used NinjaKit in the past but it no longer works for me, a lot of scripts simply won't work.

Is there such a thing as Greasemonkey for Safari?

You can use SIMBL and GreaseKit to run most Greasemonkey scripts in Safari, unmodified. Detailed installation instructions are found at this link, repeated here:

1. Download and install SIMBL
2. Quit Safari
3. Download GreaseKit
4. Drag the GreaseKit.bundle file to ~/Library/Application Support/SIMBL/Plugins. You may need to create this location if it doesn't exit
5. Start Safari and you'll see a GreaseKit menu bar item
6. Install scripts from http://userscripts.org -- the installation process is a little different from the Firefox approach, but it works.

Or would one instead write a Safari Extension to customize behavior on other web pages? Maybe there's an alternative that runs on AppleScript instead of JavaScript?

The native-to-Safari equivalent of Greasemonkey is AppleScript extensions and plugins. For a nice library of native Safari extensions check out Pimp My Safari.