How to design indexes for columns with NULL values in MySQL?
It is the NOT NULL:

CREATE TEMPORARY TABLE `myTab` (`notnul` FLOAT, `nul` FLOAT);
INSERT INTO `myTab` VALUES (1, NULL), (1, 2), (1, NULL), (1, 2), (1, NULL), (1, 2), (1, NULL), (1, 2), (1, NULL), (1, 2), (1, NULL), (1, 2);
SELECT * FROM `myTab`;


gives:

+--------+------+
| notnul | nul  |
+--------+------+
|      1 | NULL |
|      1 |    2 |
|      1 | NULL |
|      1 |    2 |
|      1 | NULL |
|      1 |    2 |
|      1 | NULL |
|      1 |    2 |
|      1 | NULL |
|      1 |    2 |
|      1 | NULL |
|      1 |    2 |
+--------+------+


Create the index:

CREATE INDEX `notnul_nul` ON `myTab` (`notnul`, `nul`);
CREATE INDEX `nul_notnul` ON `myTab` (`nul`, `notnul`);

SHOW INDEX FROM `myTab`;


gives:

+-------+------------+------------+--------------+-------------+-----------+-------------+----------+--------+------+------------+---------+---------------+
| Table | Non_unique | Key_name   | Seq_in_index | Column_name | Collation | Cardinality | Sub_part | Packed | Null | Index_type | Comment | Index_comment |
+-------+------------+------------+--------------+-------------+-----------+-------------+----------+--------+------+------------+---------+---------------+
| myTab |          1 | notnul_nul |            1 | notnul      | A         |          12 |     NULL | NULL   | YES  | BTREE      |         |               |
| myTab |          1 | notnul_nul |            2 | nul         | A         |          12 |     NULL | NULL   | YES  | BTREE      |         |               |
| myTab |          1 | nul_notnul |            1 | nul         | A         |          12 |     NULL | NULL   | YES  | BTREE      |         |               |
| myTab |          1 | nul_notnul |            2 | notnul      | A         |          12 |     NULL | NULL   | YES  | BTREE      |         |               |
+-------+------------+------------+--------------+-------------+-----------+-------------+----------+--------+------+------------+---------+---------------+


now explain the selects.
It seems that MySQL uses the index, even if You use NOT NULL:

EXPLAIN SELECT * FROM `myTab` WHERE `notnul` IS NOT NULL;
+----+-------------+-------+-------+---------------+------------+---------+------+------+--------------------------+ 
| id | select_type | table | type  | possible_keys | key        | key_len | ref  | rows | Extra                    |
+----+-------------+-------+-------+---------------+------------+---------+------+------+--------------------------+ 
|  1 | SIMPLE      | myTab | index | notnul_nul    | notnul_nul | 10      | NULL |   12 | Using where; Using index |
+----+-------------+-------+-------+---------------+------------+---------+------+------+--------------------------+


EXPLAIN SELECT * FROM `myTab` WHERE `nul` IS NOT NULL;
+----+-------------+-------+-------+---------------+------------+---------+------+------+--------------------------+
| id | select_type | table | type  | possible_keys | key        | key_len | ref  | rows | Extra                    |
+----+-------------+-------+-------+---------------+------------+---------+------+------+--------------------------+
|  1 | SIMPLE      | myTab | range | nul_notnul    | nul_notnul | 5       | NULL |    6 | Using where; Using index |
+----+-------------+-------+-------+---------------+------------+---------+------+------+--------------------------+


But, when comparing NOT NULL and NULL, it seems that MySQL preferrs other indexes when using NOT NULL. Although this does obviously not add any information. This is because MySQL interprets NOT NULL as a range as you can see in type-column. I'm not sure If there is a workaround:

EXPLAIN SELECT * FROM `myTab` WHERE `nul` IS NULL &amp;&amp; notnul=2;
+----+-------------+-------+------+-----------------------+------------+---------+-------------+------+--------------------------+
| id | select_type | table | type | possible_keys         | key        | key_len | ref         | rows | Extra                    |
+----+-------------+-------+------+-----------------------+------------+---------+-------------+------+--------------------------+
|  1 | SIMPLE      | myTab | ref  | notnul_nul,nul_notnul | notnul_nul | 10      | const,const |    1 | Using where; Using index |
+----+-------------+-------+------+-----------------------+------------+---------+-------------+------+--------------------------+


EXPLAIN SELECT * FROM `myTab` WHERE `nul` IS NOT NULL &amp;&amp; notnul=2;
+----+-------------+-------+-------+-----------------------+------------+---------+------+------+--------------------------+
| id | select_type | table | type  | possible_keys         | key        | key_len | ref  | rows | Extra                    |
+----+-------------+-------+-------+-----------------------+------------+---------+------+------+--------------------------+
|  1 | SIMPLE      | myTab | range | notnul_nul,nul_notnul | notnul_nul | 10      | NULL |    1 | Using where; Using index |
+----+-------------+-------+-------+-----------------------+------------+---------+------+------+--------------------------+


I think there could be a better implementation in MySQL, because NULL is a special value. Probably most people are interested in NOT NULL values.