register_setting() │ WP 2.7.0

An example showing how to connect the function via hook admin_init:

add_action( 'admin_init', 'register_my_setting' );

function register_my_setting() {
	register_setting( 'my_options_group', 'my_option_name', 'intval' ); 
} 

For more examples, see Options API.

If registration is done for both site and REST API, then function call should be put on both rest_api_init and admin_init hooks. If you use only admin_init hook, the option will not work in REST.

In this case the option will also be visible when accessing the wp-json/wp/v2/settings endpoint.

add_action( 'admin_init', 'register_my_setting' );
add_action( 'rest_api_init', 'register_my_setting' );

function register_my_setting() {

	$args = array(
		'sanitize_callback' => 'sanitize_text_field',
		'default'      => '',
		'type'         => 'string',
		'show_in_rest' => true,
	);

	register_setting( 'my_options_group', 'my_option_name', $args ); 
} 

add_action( 'rest_api_init', 'register_block_core_site_logo_setting', 10 );

/**
 * Register a core site setting for a site logo
 */
function register_block_core_site_logo_setting() {

	register_setting( 'general', 'site_logo', [
		'show_in_rest' => array(
			'name' => 'site_logo',
		),
		'type'         => 'integer',
		'description'  => __( 'Site logo.' ),
	] );
}

If you want to save the array in settings and also show this array in the wp-json/wp/v2/settings endpoint, you must specify the array scheme in the show_in_rest parameter.

add_action( 'admin_init',    'wpdocs_foo_register_settings' );
add_action( 'rest_api_init', 'wpdocs_foo_register_settings' );

function wpdocs_foo_register_settings() {

	register_setting( 'general_setting', 'id' );
	register_setting( 'general_setting', 'order' );
	register_setting(
		'general_setting',
		'slider-data',
		array(
			'show_in_rest' => array(
				'name' => 'images_slide',
				'schema' => array(
					'type'  => 'array',
					'items' => array(
						'id'    => 'string',
						'order' => 'string',
					),
				),
			),
			'type' => 'array',
			'sanitize_callback' => array( $this, 'wpdocs_admin_post_save_data' ),
		)
	);
}

register_setting(
	'dp_example_settings_group',
	'dp_example_array_settings',
	array(
		'type'         => 'object',
		'default'      => array(
			'A',
			'B',
			'C',
		),
		'show_in_rest' => array(
			'schema' => array(
				'type'  => 'object',
				'items' => array(
					'type' => 'string',
				),
			),
		),
	)
);

register_setting(
	'dp_example_settings_group',
	'dp_example_object_settings',
	array(
		'type'         => 'object',
		'default'      => array(
			'some_str' => 'A',
			'some_int' => 3,
		),
		'show_in_rest' => array(
			'schema' => array(
				'type'       => 'object',
				'properties' => array(
					'some_str' => array(
						'type' => 'string',
					),
					'some_int' => array(
						'type' => 'integer',
					),
				),
			),
		),
	)
);

As mentioned in the note above, sanitize can work twice when the option is first added to the database. Therefore, if there are any performance-critical points or other points that need to be taken into account in this case, they need to be handled separately in the sanitize function.

add_action( 'admin_init', 'on_admin_init' );

/*
 *  Add custom options to whitelist, allowing valiated settings to be saved by form.
 */
function on_admin_init(): void 
{
	register_setting( PLUGIN_SLUG, PLUGIN_SLUG, [ 'sanitize_callback' => 'my_sanitize_settings' ] ); 
}

/*
 *  Sanitize the form input.
 */
function my_sanitize_settings( $input = NULL ):  
{
	// Detect multiple sanitizing passes.
	// Accomodates bug: https://core.trac.wordpress.org/ticket/21989
	static $pass_count = 0; $pass_count++;

	if ( $pass_count <= 1 ) {
		// Handle any single-time / performane sensitive actions.

	}

	// Insert regular santizing code here.
}

The function only fires twice the first time a setting is saved and added to a DB table. Subsequent updates to the setting perform the sanitize_callback function only once. So unless you are doing some very, very, very heavy sanitize operation, there is no need to worry about it.