Depending on your design choices, the patterns you use (if any at all) will vary. Using Adapters and Decorators with RMI is a popular choice, and using Factory patterns with Sockets is also common.
You should definitely implement some kind of MVC pattern to separate your GUI presentation from your data access logic: it's good form and decouples the different tiers of the project.
Keep in mind that Design patterns are "tried and true" solutions to common problems faced in coding. You should use a design pattern when 1) you run into one of these problems 2) a simpler solution is not available and 3) the pattern makes sense for what you're doing. I've worked on projects where they thought it was a good idea to string together a whole bunch of design patterns because "design patterns are good!" and made the solution a lot messier and a lot more complex than it needed to be.
My advice is to keep things as simple as possible, and when you come across a problem where a well known pattern will get the job done, go ahead and implement that pattern. I wouldn't do things the other way around and say "Where can I implement those good old patterns?" before you even work towards your solution.
Hope that gives you some insight.
SCJP 1.4, SCWCD J2EE 1.4, SCJD J2SE 1.5, SCBCD J2EE 1.3, SCDJWS (In Progress)