If you were used putting your Mac into hibernate mode using the free Deep Sleep Widget you were out of luck after upgrading to OS X Mountain Lion. The widget was still able to put your system into hibernate mode but after starting your computer again all you got was 3 beeps followed by a normal reboot of your system.
As the Deep Sleep Widget is open source I had a look at it and was able to fix the widget for Mountain Lion.
Get the current version here.
Extract the archive and start the installation by double-clicking the widget.
It should run on all Intel-based Macs and I've tested this version on Snow Leopard and Mountain Lion. If you were able to run it on a different OS version it would be nice if you leave a comment here.