The optimization process and recommendations:

1. Create Optimal Indexes (modified query below): The recommended indexes are an integral part of this optimization effort and should be created before testing the execution duration of the optimized query.
2. Replace In Subquery With Correlated Exists (modified query below): In many cases, an EXISTS subquery with a correlated condition will perform better than a non correlated IN subquery.

Optimal indexes for this query:

ALTER TABLE `mailing_list` ADD INDEX `mailing_list_idx_optout` (`optout`);
ALTER TABLE `mailing_list` ADD INDEX `mailing_list_idx_email_optout` (`email`,`optout`);

The optimized query:

SELECT
        DISTINCT TRIM(mailing_list.email) AS emails 
    FROM
        mailing_list 
    WHERE
        mailing_list.optout = 0 
        AND NOT EXISTS (
            SELECT
                1 
            FROM
                mailing_list AS mailing_list1 
            WHERE
                (
                    mailing_list1.optout = 1
                ) 
                AND (
                    mailing_list.email = mailing_list1.email
                )
        ) 
    ORDER BY
        emails