How to create a jQuery plugin with methods?

Here's the pattern I have used for creating plugins with additional methods. You would use it like:

$('selector').myplugin( { key: 'value' } );

or, to invoke a method directly,

$('selector').myplugin( 'mymethod1', 'argument' );

Example:

;(function($) {

    $.fn.extend({
        myplugin: function(options,arg) {
            if (options && typeof(options) == 'object') {
                options = $.extend( {}, $.myplugin.defaults, options );
            }

            // this creates a plugin for each element in
            // the selector or runs the function once per
            // selector.  To have it do so for just the
            // first element (once), return false after
            // creating the plugin to stop the each iteration 
            this.each(function() {
                new $.myplugin(this, options, arg );
            });
            return;
        }
    });

    $.myplugin = function( elem, options, arg ) {

        if (options && typeof(options) == 'string') {
           if (options == 'mymethod1') {
               myplugin_method1( arg );
           }
           else if (options == 'mymethod2') {
               myplugin_method2( arg );
           }
           return;
        }

        ...normal plugin actions...

        function myplugin_method1(arg)
        {
            ...do method1 with this and arg
        }

        function myplugin_method2(arg)
        {
            ...do method2 with this and arg
        }

    };

    $.myplugin.defaults = {
       ...
    };

})(jQuery);

What about this approach:

jQuery.fn.messagePlugin = function(){
    var selectedObjects = this;
    return {
             saySomething : function(message){
                              $(selectedObjects).each(function(){
                                $(this).html(message);
                              });
                              return selectedObjects; // Preserve the jQuery chainability 
                            },
             anotherAction : function(){
                               //...
                               return selectedObjects;
                             }
           };
}
// Usage:
$('p').messagePlugin().saySomething('I am a Paragraph').css('color', 'red');

The selected objects are stored in the messagePlugin closure, and that function returns an object that contains the functions associated with the plugin, the in each function you can perform the desired actions to the currently selected objects.

You can test and play with the code here.

Edit: Updated code to preserve the power of the jQuery chainability.

The following plugin-structure utilizes the jQuery-data()-method to provide a public interface to internal plugin-methods/-settings (while preserving jQuery-chainability):

(function($, window, undefined) { 
  const defaults = {
    elementId   : null,
    shape       : "square",
    color       : "aqua",
    borderWidth : "10px",
    borderColor : "DarkGray"
  };

  $.fn.myPlugin = function(options) {
    // settings, e.g.:  
    var settings = $.extend({}, defaults, options);

    // private methods, e.g.:
    var setBorder = function(color, width) {        
      settings.borderColor = color;
      settings.borderWidth = width;          
      drawShape();
    };

    var drawShape = function() {         
      $('#' + settings.elementId).attr('class', settings.shape + " " + "center"); 
      $('#' + settings.elementId).css({
        'background-color': settings.color,
        'border': settings.borderWidth + ' solid ' + settings.borderColor      
      });
      $('#' + settings.elementId).html(settings.color + " " + settings.shape);            
    };

    return this.each(function() { // jQuery chainability     
      // set stuff on ini, e.g.:
      settings.elementId = $(this).attr('id'); 
      drawShape();

      // PUBLIC INTERFACE 
      // gives us stuff like: 
      //
      //    $("#...").data('myPlugin').myPublicPluginMethod();
      //
      var myPlugin = {
        element: $(this),
        // access private plugin methods, e.g.: 
        setBorder: function(color, width) {        
          setBorder(color, width);
          return this.element; // To ensure jQuery chainability 
        },
        // access plugin settings, e.g.: 
        color: function() {
          return settings.color;
        },        
        // access setting "shape" 
        shape: function() {
          return settings.shape;
        },     
        // inspect settings 
        inspectSettings: function() {
          msg = "inspecting settings for element '" + settings.elementId + "':";   
          msg += "\n--- shape: '" + settings.shape + "'";
          msg += "\n--- color: '" + settings.color + "'";
          msg += "\n--- border: '" + settings.borderWidth + ' solid ' + settings.borderColor + "'";
          return msg;
        },               
        // do stuff on element, e.g.:  
        change: function(shape, color) {        
          settings.shape = shape;
          settings.color = color;
          drawShape();   
          return this.element; // To ensure jQuery chainability 
        }
      };
      $(this).data("myPlugin", myPlugin);
    }); // return this.each 
  }; // myPlugin
}(jQuery));

Now you can call internal plugin-methods to access or modify plugin data or the relevant element using this syntax:

$("#...").data('myPlugin').myPublicPluginMethod(); 

As long as you return the current element (this) from inside your implementation of myPublicPluginMethod() jQuery-chainability will be preserved - so the following works:

$("#...").data('myPlugin').myPublicPluginMethod().css("color", "red").html("...."); 

Here are some examples (for details checkout this fiddle):

// initialize plugin on elements, e.g.:
$("#shape1").myPlugin({shape: 'square', color: 'blue', borderColor: 'SteelBlue'});
$("#shape2").myPlugin({shape: 'rectangle', color: 'red', borderColor: '#ff4d4d'});
$("#shape3").myPlugin({shape: 'circle', color: 'green', borderColor: 'LimeGreen'});

// calling plugin methods to read element specific plugin settings:
console.log($("#shape1").data('myPlugin').inspectSettings());    
console.log($("#shape2").data('myPlugin').inspectSettings());    
console.log($("#shape3").data('myPlugin').inspectSettings());      

// calling plugin methods to modify elements, e.g.:
// (OMG! And they are chainable too!) 
$("#shape1").data('myPlugin').change("circle", "green").fadeOut(2000).fadeIn(2000);      
$("#shape1").data('myPlugin').setBorder('LimeGreen', '30px');

$("#shape2").data('myPlugin').change("rectangle", "red"); 
$("#shape2").data('myPlugin').setBorder('#ff4d4d', '40px').css({
  'width': '350px',
  'font-size': '2em' 
}).slideUp(2000).slideDown(2000);              

$("#shape3").data('myPlugin').change("square", "blue").fadeOut(2000).fadeIn(2000);   
$("#shape3").data('myPlugin').setBorder('SteelBlue', '30px');

// etc. ...