Just a rough plugin example to answer a WPSE question: How to display Wordpress User Registration Form in front-end of the website?

Custom_Reg.php

<?php 
namespace Custom_Reg;

class Custom_Reg {

  protected $form;
  
  protected $saver;
  
  function __construct( Form $form, Saver $saver ) {
    $this->form = $form;
    $this->saver = $saver;
  }
  
  /**
   * Check if the url to recognize is the one for the registration form page
   */
  function checkUrl() {
    $url_part = $this->getUrl();
    $nonce = urlencode( wp_create_nonce( 'registration_url' ) );
    if ( ( $url_part === $nonce ) ) {
      // do nothing if registration is not allowed or user logged
      if ( is_user_logged_in() || ! get_option('users_can_register') ) {
        wp_safe_redirect( home_url() );
        exit();
      }
      return TRUE;
    }
  }
  
  /**
   * Init the form, if submitted validate and save, if not just display it
   */
  function init() {
    if ( $this->checkUrl() !== TRUE ) return;
    do_action( 'custom_reg_form_init', $this->form );
    if ( $this->isSubmitted() ) {
      $this->save();
    }
    // don't need to create form if already saved
    if ( ! isset( $custom_reg_form_done ) || ! $custom_reg_form_done ) {
      $this->form->create();
    }
    load_template( $this->getTemplate() );
    exit();
  }
  
  protected function save() {
	global $custom_reg_form_error;
	$this->saver->setFields( $this->form->getFields() );
	if ( $this->saver->validate() === TRUE ) { // validate?
	  if ( $this->saver->save() ) { // saved?
		global $custom_reg_form_done;
		$custom_reg_form_done = TRUE;
	  } else { // saving error
		$err =  $this->saver->getErrorMessage(); 
		$custom_reg_form_error = $err ? : __( 'Error on save.', 'custom_reg_form' );
	  }
	} else { // validation error
	   $custom_reg_form_error = $this->saver->getErrorMessage();
	}
  }
  
  protected function isSubmitted() {
    $type = $this->form->getVerb() === 'GET' ? INPUT_GET : INPUT_POST;
    $sub = filter_input( $type, 'custom_reg_form', FILTER_SANITIZE_STRING );
    return ( ! empty( $sub ) && $sub === get_class( $this->form ) );
  }
  
  protected function getTemplate() {
    $base = $this->form->getTemplate() ? : FALSE;
    $template = FALSE;
    $default = dirname( __FILE__ ) . '/default_form_template.php';
    if ( ! empty( $base ) ) {
      $template = locate_template( $base );
    }
    return $template ? : $default;
  }
  
   protected function getUrl() {
    $home_path = trim( parse_url( home_url(), PHP_URL_PATH ), '/' );
    $relative = trim( str_replace( $home_path, '', add_query_arg( array() ) ), '/' );
    $parts = explode( '/', $relative );
    if ( ! empty( $parts ) && ! isset( $parts[1] ) ) {
      return $parts[0];
    }
  }
}

custom_registration_form.php

<?php 
/**
 * Plugin Name: Custom Registration Form
 * Description: Just a rough plugin example to answer a WPSE question
 * Plugin URI: http://wordpress.stackexchange.com/questions/10309/
 * Author: G. M.
 * Author URI: http://wordpress.stackexchange.com/users/35541/g-m
 *
 */
 
if ( is_admin() ) return;

load_plugin_textdomain(
  'custom_reg_form',
  FALSE,
  plugin_dir_path( __FILE__ ) . 'langs'
); 

require_once plugin_dir_path( __FILE__ ) . 'FieldInterface.php';
require_once plugin_dir_path( __FILE__ ) . 'fields.php';
require_once plugin_dir_path( __FILE__ ) . 'Form.php';
require_once plugin_dir_path( __FILE__ ) . 'Saver.php';
require_once plugin_dir_path( __FILE__ ) . 'CustomReg.php';



/**
* Generate dynamic registration url
*/
function custom_registration_url() {
  $nonce = urlencode( wp_create_nonce( 'registration_url' ) );
  return home_url( $nonce );
}

/**
* Generate dynamic registration link
*/
function custom_registration_link() {
  $format = '<a href="%s">%s</a>';
  printf(
    $format,
    custom_registration_url(), __( 'Register', 'custom_reg_form' )
  );
}

/**
* Setup, show and save the form
*/
add_action( 'wp_loaded', function() {
  try {
    $form = new Custom_Reg\Form;
    $saver = new Custom_Reg\Saver;
    $custom_reg = new Custom_Reg\Custom_Reg( $form, $saver );
    $custom_reg->init();
  } catch ( Exception $e ) {
    if ( defined('WP_DEBUG') && WP_DEBUG ) {
      $msg = 'Exception on  ' . __FUNCTION__;
      $msg .= ', Type: ' . get_class( $e ) . ', Message: ';
      $msg .= $e->getMessage() ? : 'Unknown error';
      error_log( $msg );
    }
    wp_safe_redirect( home_url() );
  }
}, 0 );

/**
* Add fields to form
*/
add_action( 'custom_reg_form_init', function( $form ) {
  $classes = array(
    'Custom_Reg\FullName',
    'Custom_Reg\Login',
    'Custom_Reg\Email',
    'Custom_Reg\Country'
  );
  foreach ( $classes as $class ) {
    $form->addField( new $class );
  }
}, 1 );

default_form_template.php

<?php
get_header();

global $custom_reg_form_done, $custom_reg_form_error;

if ( isset( $custom_reg_form_done ) && $custom_reg_form_done ) {
  echo '<p class="success">';
  _e(
	  'Thank you, your registration was submitted, check your email.',
	  'custom_reg_form'
  );
  echo '</p>';
} else {
  if ( $custom_reg_form_error ) {
	  echo '<p class="error">' . $custom_reg_form_error  . '</p>';
  }
  do_action( 'custom_registration_form' );
}

get_footer();

FieldInterface.php

<?php 
namespace Custom_Reg;

interface FieldInterface {
  
  /**
   * Return the field id, used to name the request value and for the 'name' param of
   * html input field
   */
  public function getId();

  /**
   * Return the filter constant that must be used with
   * filter_input so get the value from request
   */
  public function getFilter();

  /**
   * Return true if the used value passed as argument should be accepted, false if not
   */
  public function isValid( $value = NULL );

  /**
   * Return true if field is required, false if not
   */
  public function isRequired();

  /**
   * Return the field input markup. The 'name' param must be output 
   * according to getId()
   */
  public function create( $value = '');

fields.php

<?php
namespace Custom_Reg;

abstract class BaseField implements FieldInterface {

  protected function getType() {
    return isset( $this->type ) ? $this->type : 'text';
  }
  
  protected function getClass() {
    $type = $this->getType();
    if ( ! empty($type) ) return "{$type}-field";
  }
  
  public function getFilter() {
    return FILTER_SANITIZE_STRING;
  }
  
  public function isRequired() {
    return isset( $this->required ) ? $this->required : FALSE;
  }
  
  public function isValid( $value = NULL ) {
    if ( $this->isRequired() ) {
      return $value != '';
    }
    return TRUE;
  }
  
  public function create( $value = '' ) {
    $label = '<p><label>' . $this->getLabel() . '</label>';
    $format = '<input type="%s" name="%s" value="%s" class="%s"%s /></p>';
    $required = $this->isRequired() ? ' required' : '';
    return $label . sprintf(
      $format,
      $this->getType(), $this->getId(), $value, $this->getClass(), $required
    );
  }

  abstract function getLabel();
}


class FullName extends BaseField {

  protected $required = TRUE;
  
  public function getID() {
    return 'fullname';
  }
  
  public function getLabel() {
    return __( 'Full Name', 'custom_reg_form' );
  }

}

class Login extends BaseField {

  protected $required = TRUE;

  public function getID() {
    return 'login';
  }

  public function getLabel() {
    return __( 'Username', 'custom_reg_form' );
  }
}

class Email extends BaseField {

  protected $type = 'email';

  public function getID() {
    return 'email';
  }

  public function getLabel() {
    return __( 'Email', 'custom_reg_form' );
  }

  public function isValid( $value = NULL ) {
    return ! empty( $value ) && filter_var( $value, FILTER_VALIDATE_EMAIL );
  }
}

class Country extends BaseField {

  protected $required = FALSE;

  public function getID() {
    return 'country';
  }

  public function getLabel() {
    return __( 'Country', 'custom_reg_form' );
  }
}

Form.php

<?php 
namespace Custom_Reg;

class Form {
  
  protected $fields;
  
  protected $verb = 'POST';
  
  protected $template;
  
  protected $form;
  
  public function __construct() {
    $this->fields = new \ArrayIterator();
  }
  
  public function create() {
    do_action( 'custom_reg_form_create', $this );
    $form = $this->open();
    $it =  $this->getFields();
    $it->rewind();
    while( $it->valid() ) {
      $field = $it->current();
      if ( ! $field instanceof FieldInterface ) {
        throw new \DomainException( "Invalid field" );
      }
      $form .= $field->create() . PHP_EOL;
      $it->next();
    }
    do_action( 'custom_reg_form_after_fields', $this );
    $form .= $this->close();
    $this->form = $form;
    add_action( 'custom_registration_form', array( $this, 'output' ), 0 );
  }
  
  public function output() {
    unset( $GLOBALS['wp_filters']['custom_reg_form'] );
    if ( ! empty( $this->form ) ) {
      echo $this->form;
    }
  }
  
  public function getTemplate() {
    return $this->template;
  }
  
  public function setTemplate( $template ) {
    if ( ! is_string( $template ) ) {
      throw new \InvalidArgumentException( "Invalid template" );
    }
    $this->template = $template;
  }
  
  public function addField( FieldInterface $field ) {
    $hook = 'custom_reg_form_create';
    if ( did_action( $hook ) && current_filter() !== $hook ) {
      throw new \BadMethodCallException( "Add fields before {$hook} is fired" );
    }
    $this->getFields()->append( $field );
  }
  
  public function getFields() {
    return $this->fields;
  }
  
  public function getVerb() {
    return $this->verb;
  }
  
  public function setVerb( $verb ) {
    if ( ! is_string( $verb) ) {
     throw new \InvalidArgumentException( "Invalid verb" );
    }
    $verb = strtoupper($verb);
    if ( in_array($verb, array( 'GET', 'POST' ) ) ) $this->verb = $verb;
  }
  
  protected function open() {
    $out = sprintf( '<form id="custom_reg_form" method="%s">', $this->verb ) . PHP_EOL;
    $nonce = '<input type="hidden" name="_n" value="%s" />';
    $out .= sprintf( $nonce,  wp_create_nonce( 'custom_reg_form_nonce' ) ) . PHP_EOL;
    $identity = '<input type="hidden" name="custom_reg_form" value="%s" />';
    $out .= sprintf( $identity,  __CLASS__ ) . PHP_EOL;
    return $out;
  }
  
  protected function close() {
    $submit =  __('Register', 'custom_reg_form');
    $out = sprintf( '<input type="submit" value="%s" />', $submit );
    $out .= '</form>';
    return $out;
  }

}

Saver.php

<?php
namespace Custom_Reg;

class Saver {
    
  protected $fields;
  
  protected $user = array( 'user_login' => NULL, 'user_email' => NULL );
  
  protected $meta = array();
  
  protected $error;
  
  public function setFields( \ArrayIterator $fields ) {
    $this->fields = $fields;
  }
  
  /**
  * validate all the fields
  */
  public function validate() {
    // if registration is not allowed return false
    if ( ! get_option('users_can_register') ) return FALSE;
    // if registration is not allowed return false
    if ( ! $this->getFields() instanceof \ArrayIterator ) return FALSE;
    // first check nonce
    $nonce = $this->getValue( '_n' );
    if ( $nonce !== wp_create_nonce( 'custom_reg_form_nonce' ) ) return FALSE;
    // then check all fields
    $it =  $this->getFields();
    while( $it->valid() ) {
      $field = $it->current();
      $key = $field->getID();
      if ( ! $field instanceof FieldInterface ) {
        throw new \DomainException( "Invalid field" );
      }
      $value = $this->getValue( $key, $field->getFilter() );
      if ( $field->isRequired() && empty($value) ) {
        $this->error = sprintf( __('%s is required', 'custom_reg_form' ), $key );
        return FALSE;
      }
      if ( ! $field->isValid( $value ) ) {
        $this->error = sprintf( __('%s is not valid', 'custom_reg_form' ), $key );
        return FALSE;
      }
      if ( in_array( "user_{$key}", array_keys($this->user) ) ) {
        $this->user["user_{$key}"] = $value;
      } else {
        $this->meta[$key] = $value;
      }
      $it->next();
    }
    return TRUE;
  }
  
  /**
  * Save the user using core register_new_user that handle username and email check
  * and also sending email to new user
  * in addition save all other custom data in user meta
  *
  * @see register_new_user()
  */
  public function save() {
    // if registration is not allowed return false
    if ( ! get_option('users_can_register') ) return FALSE;
    // check mandatory fields
    if ( ! isset($this->user['user_login']) || ! isset($this->user['user_email']) ) {
      return false;
    }
    $user = register_new_user( $this->user['user_login'], $this->user['user_email'] );
    if ( is_numeric($user) ) {
      if ( ! update_user_meta( $user, 'custom_data', $this->meta ) ) {
        wp_delete_user($user);
        return FALSE;
      }
      return TRUE;
    } elseif ( is_wp_error( $user ) ) {
      $this->error = $user->get_error_message();
    }
    return FALSE;
  }
  
  public function getValue( $var, $filter = FILTER_SANITIZE_STRING ) {
    if ( ! is_string($var) ) {
      throw new \InvalidArgumentException( "Invalid value" );
    }
    $method = strtoupper( filter_input( INPUT_SERVER, 'REQUEST_METHOD' ) );
    $type = $method === 'GET' ? INPUT_GET : INPUT_POST;
    $val = filter_input( $type, $var, $filter );
    return $val;
  }
  
  public function getFields() {
    return $this->fields;
  }
  
  public function getErrorMessage() {
    return $this->error;
  }

}