There are loads of different design patterns used, but there are a few obvious ones:
Proxy - used heavily in AOP, and remoting.
Singleton - beans defined in spring config files are singletons by default.
Template method - used extensively to deal with boilerplate repeated code (such as closing connections cleanly, etc..). For example JdbcTemplate, JmsTemplate, JpaTemplate.
Update following comments: For MVC, you might want to read the MVC Reference
Some obvious patterns in use in MVC:
Model View Controller :-) . The advantage with Spring MVC is that your controllers are POJOs as opposed to being servlets. This makes for easier testing of controllers. One thing to note is that the controller is only required to return a logical view name, and the view selection is left to a separate ViewResolver. This makes it easier to reuse controllers for different view technologies.
Front Controller. Spring provides DispatcherServlet to ensure an incoming request gets dispatched to your controllers.
View Helper - Spring has a number of custom JSP tags, and velocity macros, to assist in separating code from presentation in views.