The <f:ajax> requires jsf.js file being included in the HTML <head>. It contains all JS functions for doing the JSF ajax magic.
To achieve this, ensure that you're using <h:head> instead of <head> in the master template. JSF will then automatically include the necessary <script> element there pointing to jsf.js.
<!DOCTYPE html>
<html lang="en"
xmlns="http://www.w3.org/1999/xhtml"
xmlns:f="http://java.sun.com/jsf/core"
xmlns:h="http://java.sun.com/jsf/html"
xmlns:ui="http://java.sun.com/jsf/facelets">
<h:head>
<title>Look, with h:head</title>
</h:head>
<h:body>
Put your content here.
</h:body>
</html>
Note that in a bit decent webbrowser with a bit decent webdeveloper toolset like Firefox's Web Developer Toolbar and/or Firebug you should immediately have noticed JS errors like jsf is undefined when the ajax request is to be executed. That should at least have given something to think about.
Update: as per your update
I've found out a few interesting things:
<f:ajax> tag doesn't work at <h:link>, <h:selectOneMenu>, <h:button>, <h:commandButton>. In this cases incorrect values in render attribute is not noticed, but incorrect value of event attribute generate an error.
<h:outputLabel>, <h:inputText> work with <f:ajax> properly.
The <h:link> and <h:button> are intented for GET requests only, not POST requests. It should however work just fine on <h:selectOneMenu> and <h:commandButton>. Don't you have more code into the complete picture which you omitted from the question for simplicity? Which JSF impl/version are you using? Are you using the right libraries in classpath? It look like that you must really have messed up something.
To convince you (and myself) I just created the following copy'n'paste'n'runnable testcase
<!DOCTYPE html>
<html lang="en"
xmlns:h="http://java.sun.com/jsf/html"
xmlns:f="http://java.sun.com/jsf/core"
xmlns:ui="http://java.sun.com/jsf/facelets"
>
<h:head>
<title>SO question 6089924</title>
</h:head>
<h:body>
<h:form>
<h:selectOneMenu value="#{bean.selected}">
<f:selectItem itemValue="#{null}" itemLabel="Select..." />
<f:selectItem itemValue="one" />
<f:selectItem itemValue="two" />
<f:selectItem itemValue="three" />
<f:ajax listener="#{bean.listener}" render="result" />
</h:selectOneMenu>
<h:commandButton value="commandButton" action="#{bean.submit}">
<f:ajax listener="#{bean.listener}" render="result" />
</h:commandButton>
<h:outputText id="result" value="#{bean.selected} #{bean.result}" />
<h:messages />
</h:form>
</h:body>
</html>
with this bean
package com.example;
import java.io.Serializable;
import javax.faces.bean.ManagedBean;
import javax.faces.bean.ViewScoped;
import javax.faces.event.AjaxBehaviorEvent;
@ManagedBean
@ViewScoped
public class Bean implements Serializable {
private String selected;
private String result;
public void submit() {
System.out.println("submit");
}
public void listener(AjaxBehaviorEvent event) {
System.out.println("listener");
result = "called by " + event.getComponent().getClass().getName();
}
public String getSelected() {
return selected;
}
public void setSelected(String selected) {
this.selected = selected;
}
public String getResult() {
return result;
}
}
It runs fine with Mojarra 2.1.1 on Tomcat 7.0.12.
INFO: Starting Servlet Engine: Apache Tomcat/7.0.12
INFO: Initializing Mojarra 2.1.1 (FCS 20110408) for context '/playground'
