Examples of GoF Design Patterns in Java's core ...

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You can find an overview of a lot design patterns in Wikipedia. It also mentions which patterns are mentioned by GoF. I'll sum them up here and try to assign as much as possible pattern implementations found in both the Java SE and Java EE API's.

Creational patterns

Abstract factory (recognizeable by creational methods returning an abstract/interface type)

Builder (recognizeable by creational methods returning the instance itself)

java.lang.StringBuilder#append() (unsynchronized)
java.lang.StringBuffer#append() (synchronized)
java.nio.ByteBuffer#put() (also on CharBuffer, ShortBuffer, IntBuffer, LongBuffer, FloatBuffer and DoubleBuffer)
javax.swing.GroupLayout.Group#addComponent()
All implementations of java.lang.Appendable

Factory method (recognizeable by creational methods returning a concrete type)

java.lang.Object#toString() (overrideable in all subclasses)
java.lang.Class#newInstance()
java.lang.Integer#valueOf(String) (also on Boolean, Byte, Character, Short, Long, Float and Double)
java.lang.Class#forName()
java.lang.reflect.Array#newInstance()
java.lang.reflect.Constructor#newInstance()

Prototype (recognizeable by creational methods returning a different instance of itself with the same properties)

java.lang.Object#clone() (the class has to implement java.lang.Cloneable)

Singleton (recognizeable by creational methods returning the same instance (usually of itself) everytime)

Structural patterns

Adapter (recognizeable by creational methods taking an instance of different abstract/interface type and returning an implementation of own/another abstract/interface type which decorates/overrides the given instance)

java.io.InputStreamReader(InputStream) (returns a Reader)
java.io.OutputStreamWriter(OutputStream) (returns a Writer)
javax.xml.bind.annotation.adapters.XmlAdapter#marshal() and #unmarshal()

Bridge (recognizeable by creational methods taking an instance of different abstract/interface type and returning an implementation of own abstract/interface type which delegates/uses the given instance)

None comes to mind yet. A fictive example would be new LinkedHashMap(LinkedHashSet<K>, List<V>) which returns an unmodifiable linked map which doesn't clone the items, but uses them. The java.util.Collections#newSetFromMap() and singletonXXX() methods however comes close.

Composite (recognizeable by behavioral methods taking an instance of same abstract/interface type into a tree structure)

java.awt.Container#add(Component) (practically all over Swing thus)
javax.faces.component.UIComponent#getChildren() (practically all over JSF UI thus)

Decorator (recognizeable by creational methods taking an instance of same abstract/interface type which adds additional behaviour)

All subclasses of java.io.InputStream, OutputStream, Reader and Writer have a constructor taking an instance of same type.
java.util.Collections, the checkedXXX(), synchronizedXXX() and unmodifiableXXX() methods.
javax.servlet.http.HttpServletRequestWrapper and HttpServletResponseWrapper

Facade (recognizeable by behavioral methods which internally uses instances of different independent abstract/interface types)

javax.faces.context.FacesContext, it internally uses among others the abstract/interface types LifeCycle, ViewHandler, NavigationHandler and many more without that the enduser has to worry about it (which are however overrideable by injection).
javax.faces.context.ExternalContext, which internally uses ServletContext, HttpSession, HttpServletRequest, HttpServletResponse, etc.

Flyweight (recognizeable by creational methods returning a cached instance, a bit the "multiton" idea)

java.lang.Integer#valueOf(int) (also on Boolean, Byte, Character, Short, Long, Float and Double)

Proxy (recognizeable by creational methods which returns an implementation of given abstract/interface type which in turn delegates/uses a different implementation of given abstract/interface type)

java.lang.reflect.Proxy
java.rmi.*, the whole API actually.

The Wikipedia example is IMHO a bit poor, lazy loading has actually completely nothing to do with the proxy pattern at all.

Behavioral patterns

Chain of responsibility (recognizeable by behavioral methods which (indirectly) invokes the same method in another implementation of same abstract/interface type in a queue)

Command (recognizeable by behavioral methods in an abstract/interface type which invokes a method in an implementation of a different abstract/interface type which has been encapsulated by the command implementation during its creation)

All implementations of java.lang.Runnable
All implementations of javax.swing.Action

Interpreter (recognizeable by behavioral methods returning a structurally different instance/type of the given instance/type; note that parsing/formatting is not part of the pattern, determining the pattern and how to apply it is)

Iterator (recognizeable by behavioral methods sequentially returning instances of a different type from a queue)

All implementations of java.util.Iterator (thus among others also java.util.Scanner!).
All implementations of java.util.Enumeration

Mediator (recognizeable by behavioral methods taking an instance of different abstract/interface type (usually using the command pattern) which delegates/uses the given instance)

java.util.Timer (all scheduleXXX() methods)
java.util.concurrent.Executor#execute()
java.util.concurrent.ExecutorService (the invokeXXX() and submit() methods)
java.util.concurrent.ScheduledExecutorService (all scheduleXXX() methods)
java.lang.reflect.Method#invoke()

Memento (recognizeable by behavioral methods which internally changes the state of the whole instance)

java.util.Date (the setter methods do that, Date is internally represented by a long value)
All implementations of java.io.Serializable
All implementations of javax.faces.component.StateHolder

Observer (or Publish/Subscribe) (recognizeable by behavioral methods which invokes a method on an instance of another abstract/interface type, depending on own state)

java.util.Observer/java.util.Observable (rarely used in real world though)
All implementations of java.util.EventListener (practically all over Swing thus)
javax.servlet.http.HttpSessionBindingListener
javax.servlet.http.HttpSessionAttributeListener
javax.faces.event.PhaseListener

State (recognizeable by behavioral methods which changes its behaviour depending on the instance's state which can be controlled externally)

javax.faces.lifecycle.LifeCycle#execute() (controlled by FacesServlet, the behaviour is dependent on current phase (state) of JSF lifecycle)

Strategy (recognizeable by behavioral methods in an abstract/interface type which invokes a method in an implementation of a different abstract/interface type which has been passed-in as method argument into the strategy implementation)

java.util.Comparator#compare(), executed by among others Collections#sort().
javax.servlet.http.HttpServlet, the service() and all doXXX() methods take HttpServletRequest and HttpServletResponse and the implementor has to process them (and not to get hold of them as instance variables!).
javax.servlet.Filter#doFilter()

Template method (recognizeable by behavioral methods which already have a "default" behaviour definied by an abstract type)

All non-abstract methods of java.io.InputStream, java.io.OutputStream, java.io.Reader and java.io.Writer.
All non-abstract methods of java.util.AbstractList, java.util.AbstractSet and java.util.AbstractMap.
javax.servlet.http.HttpServlet, all the doXXX() methods by default sends a HTTP 405 "Method Not Allowed" error to the response. You're free to implement none or any of them.

Visitor (recognizeable by two different abstract/interface types which has methods definied which takes each the other abstract/interface type; the one actually calls the method of the other and the other executes the desired strategy on it)