No matter how dates are stored in the operating system itself, the applications can do whatever they want, including resorting to the original date trick that started this Y2K mess, or inventing their own bizarre system for tracking time and dates. First, you should make an inventory of your system and all the software on it, and make sure you include the version number for each program. Then you should use the web to find the home pages for the various companies and developers who created your software, and see what they say about Y2K compliance. For Linux applications, you'll almost always get good news, and can simply check each program off your list. One simple way to test is to simply set the clock ahead on your system to sometime in January of 2000 and run your system.
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As you test, look for how programs read stored data, such as spreadsheets or financial information, how they handle displaying dates and reading them from your typed input, and how programs interact, if you feed the output of one program to another. These are the areas that are most likely to cause problems. In some cases, a program will let you enter a year as either two or four digits, and you can avoid problems by using the four-digit version. If you find a problem with a program, your first recourse is to contact the creator for a remedy. If that fails, then you have to find a replacement program, decide to do without that program entirely, or find a way to run with your computer's system clock set back a few years until you can find a more permanent solution.