Allow users to login to the site with their username or email.

wp_allow_user_login_with_email.php

<?php
/**
 *  Allow users to login to the site with their username or email.
 *
 *  @param   object  $user            WP_User object
 *  @param   string  $username_email  The username or email entered for login
 *  @param   string  $password        The users password
 *  @return  object                   The authenticated WP_User object
 */
function wp_allow_user_login_with_email( $user, $username_email, $password ) {
    // Check for empty fields
    if( empty( $username_email ) || empty( $password ) ) {
        $error = new WP_Error();

        if( empty( $username_email ) )
            $error->add( 'empty_username', __( '<strong>ERROR</strong>: The username/email field is empty.' ) );

        if( empty( $password ) )
            $error->add( 'empty_password', __( '<strong>ERROR</strong>: The password field is empty.' ) );

        return $error;
    }

    // Check if user exists in WordPress database
    if ( ! $user = get_user_by( 'login', $username_email ) ) {
        if( ! $user = get_user_by( 'email', $username_email ) ) {
            return new WP_Error( 'invalid_username', sprintf( __( '<strong>ERROR</strong>: Invalid username/email. <a href="%s">Lost your password?</a>' ), wp_lostpassword_url() ) );
        }
    }

    // Filter whether the given user can be authenticated with the provided $password.
    $user = apply_filters( 'wp_authenticate_user', $user, $password );
    if ( is_wp_error( $user ) )
        return $user;

    // Check if the password used is valid for the user
    if ( ! wp_check_password( $password, $user->user_pass, $user->ID ) )
        return new WP_Error( 'incorrect_password', sprintf( __( '<strong>ERROR</strong>: The password you entered for the username/email <strong>%1$s</strong> is incorrect. <a href="%2$s">Lost your password?</a>' ), $username_email, wp_lostpassword_url() ) );

    return $user;
}
remove_filter( 'authenticate', 'wp_authenticate_username_password', 20 );
add_filter( 'authenticate', 'allow_user_login_with_email', 20, 3 );

/**
 *  Modify the label of the username field on the login page
 *
 *  @param   string  $translated_text  The new text
 *  @param   string  $text             The current text
 *  @param   string  $domain           The sites domain name
 *  @return  string                    The new text
 */
function modify_username_label( $translated_text, $text, $domain ) {
    if ($text === 'Username') {
        $translated_text = 'Username/Email';
    }

    return $translated_text;
}
add_action( 'login_head', function() {
    add_filter( 'gettext', 'modify_username_label', 20, 3 );
});