To enable and use hibernation with Xubuntu, do the following:
Install the 
pm-utils package
from GNOME
Software.
From the command line, enter: sudo pm-hibernate.
Enter your password.
To resume from hibernation, press the power button.
I get strange patterns on the screen when I hibernate my computer
Your screen may show a black and white pattern just after you click to hibernate your computer. This is usually nothing to worry about and is just how the graphics cards of some computers respond to the initial stages of the hibernation process. If the computer displays the pattern for a prolonged period of time without turning itself off then you may have a problem with hibernation. See My computer doesn't suspend or hibernate correctly for more information.