Mrthdrb

how can I get products where the value of a multiselect is not set ?

I've got these so far :

$filter =
 array(
 array('eq' => array("378")), //multiselect with one value
 array('eq' => array("378,379")), //multiselect with 2 values
 array('eq' => false), //multiselect with no value, need to make this work
);

$collection->addAttributeToFilter( "multiselect", $filter );

Try this:

$filter =
 array(
 array('eq' => array("378")), // multiselect with one value
 array('eq' => array("378,379")), // multiselect with 2 values
 array('null' => true), // multiselect with no value (null value)
);

$collection->addAttributeToFilter( "multiselect", $filter );

Update:

here is how it will work using the SearchCriteria with the product repository (tested on the custom attribute available_shipping_methods having a multiselect input type):

use MagentoCatalogApiProductRepositoryInterface;
use MagentoFrameworkApiFilterBuilder;
use MagentoFrameworkApiSearchFilterGroupBuilder;
use MagentoFrameworkApiSearchCriteriaBuilderFactory;
use MagentoFrameworkAppActionAction;
use MagentoFrameworkAppActionContext;
use MagentoFrameworkAppResponseInterface;

/**
 * Class Search
 */
class Search extends Action

 /**
 * @var ProductRepositoryInterface
 */
 private $productRepository;

 /**
 * @var SearchCriteriaBuilderFactory
 */
 private $searchCriteriaBuilderFactory;

 /**
 * @var FilterBuilder
 */
 private $filterBuilder;

 /**
 * @var FilterGroupBuilder
 */
 private $filterGroupBuilder;

 /**
 * Search constructor.
 *
 * @param Context $context
 * @param ProductRepositoryInterface $productRepository
 * @param SearchCriteriaBuilderFactory $searchCriteriaBuilderFactory
 * @param FilterBuilder $filterBuilder
 * @param FilterGroupBuilder $filterGroupBuilder
 */
 public function __construct(
 Context $context,
 ProductRepositoryInterface $productRepository,
 SearchCriteriaBuilderFactory $searchCriteriaBuilderFactory,
 FilterBuilder $filterBuilder,
 FilterGroupBuilder $filterGroupBuilder
 ) 
 parent::__construct($context);
 $this->productRepository = $productRepository;
 $this->searchCriteriaBuilderFactory = $searchCriteriaBuilderFactory;
 $this->filterBuilder = $filterBuilder;
 $this->filterGroupBuilder = $filterGroupBuilder;
 

 /**
 * Execute action based on request and return result
 *
 * Note: Request will be added as operation argument in future
 *
 * @return MagentoFrameworkControllerResultInterface

Result query looks like this (our conditions at the last line):

SELECT `e`.*, IF(at_status.value_id > 0, at_status.value, at_status_default.value) AS `status`, IF(at_visibility.value_id > 0, at_visibility.value, at_visibility_default.value) AS `visibility`, IF(at_available_shipping_methods.value_id > 0, at_available_shipping_methods.value, at_available_shipping_methods_default.value) AS `available_shipping_methods`, `stock_status_index`.`stock_status` AS `is_salable` FROM `catalog_product_entity` AS `e`
 INNER JOIN `catalog_product_entity_int` AS `at_status_default` ON (`at_status_default`.`row_id` = `e`.`row_id`) AND (`at_status_default`.`attribute_id` = '97') AND `at_status_default`.`store_id` = 0
 LEFT JOIN `catalog_product_entity_int` AS `at_status` ON (`at_status`.`row_id` = `e`.`row_id`) AND (`at_status`.`attribute_id` = '97') AND (`at_status`.`store_id` = 1)
 INNER JOIN `catalog_product_entity_int` AS `at_visibility_default` ON (`at_visibility_default`.`row_id` = `e`.`row_id`) AND (`at_visibility_default`.`attribute_id` = '99') AND `at_visibility_default`.`store_id` = 0
 LEFT JOIN `catalog_product_entity_int` AS `at_visibility` ON (`at_visibility`.`row_id` = `e`.`row_id`) AND (`at_visibility`.`attribute_id` = '99') AND (`at_visibility`.`store_id` = 1)
 INNER JOIN `catalog_product_entity_text` AS `at_available_shipping_methods_default` ON (`at_available_shipping_methods_default`.`row_id` = `e`.`row_id`) AND (`at_available_shipping_methods_default`.`attribute_id` = '265') AND `at_available_shipping_methods_default`.`store_id` = 0
 LEFT JOIN `catalog_product_entity_text` AS `at_available_shipping_methods` ON (`at_available_shipping_methods`.`row_id` = `e`.`row_id`) AND (`at_available_shipping_methods`.`attribute_id` = '265') AND (`at_available_shipping_methods`.`store_id` = 1)
 INNER JOIN `cataloginventory_stock_status` AS `stock_status_index` ON e.entity_id = stock_status_index.product_id WHERE (((IF(at_available_shipping_methods.value_id > 0, at_available_shipping_methods.value, at_available_shipping_methods_default.value) LIKE '%shipping_shipping%') OR (IF(at_available_shipping_methods.value_id > 0, at_available_shipping_methods.value, at_available_shipping_methods_default.value) LIKE '%flatrate_flatrate%') OR (IF(at_available_shipping_methods.value_id > 0, at_available_shipping_methods.value, at_available_shipping_methods_default.value) IS NULL))) AND (e.created_in <= 1) AND (e.updated_in > 1)

and result itself contains products with desired attribute value and without value at all (null).

PS: a comma-separated value of the multiselect attribute is the main problem, because of that we need to use %LIKE% condition, what is not a good idea.